Construction of smooth isomorphic and finite-to-one extensions of irrational rotations which are not almost automorphic

Abstract

Due to a result by Glasner and Downarowicz [Isomorphic extensions and applications. Topol. Methods Nonlinear Anal. 48(1) (2016), 321–338], it is known that a minimal system is mean equicontinuous if and only if it is an isomorphic extension of its maximal equicontinuous factor. The majority of known examples of this type are almost automorphic, that is, the factor map to the maximal equicontinuous factor is almost one-to-one. The only cases of isomorphic extensions which are not almost automorphic are again due to Glasner and Downarowicz, who in the same article provide a construction of such systems in a rather general topological setting. Here, we use the Anosov–Katok method to provide an alternative route to such examples and to show that these may be realized as smooth skew product diffeomorphisms of the two-torus with an irrational rotation on the base. Moreover – and more importantly – a modification of the construction allows to ensure that lifts of these diffeomorphisms to finite covering spaces provide novel examples of finite-to-one topomorphic extensions of irrational rotations. These are still strictly ergodic and share the same dynamical eigenvalues as the original system, but show an additional singular continuous component of the dynamical spectrum.

1. Introduction

The celebrated Halmos–von Neumann theorem provides a classification, up to isomorphism, of ergodic measure-preserving dynamical systems with discrete dynamical spectrum. Moreover, any such system can be realized as a rotation on some compact abelian group [HvN42, vN32]. From the measure-theoretic viewpoint, this provides a rather complete picture for the class of dynamical systems with discrete spectrum. However, topological realizations of such systems can still show a surprising variety of different behaviours. One particular subclass that has recently attracted considerable attention are mean equicontinuous systems [GRLZ19, GRM19, HLY11, LTY15, LYY21]. In the minimal case, Downarowicz and Glasner showed that these are exactly those topological dynamical systems which are measure-theoretically isomorphic to their maximal equicontinuous factor (MEF) via the respective continuous factor map [DG16]. Such systems are called isomorphic extensions (of the MEF). Equivalently, these systems are characterized by discrete spectrum with continuous eigenfunctions. A generalization of these results to the non-minimal case and more general group actions is provided in [FGL22]. Subsequent work has concentrated on characterizing different types of mean equicontinuous systems in terms of invertibility properties of the factor map to the MEF. For instance, the fact that almost all points of a strictly ergodic system are injectivity points of the factor map, which implies mean equicontinuity, is equivalent to the stronger property of diam-mean equicontinuity [GR17, GRJY21].

Examples of mean equicontinuous systems in the literature are abundant. In particular, these include the classes of regular Toeplitz flows [Dow05, JK69, MP79, Wil84] and regular model sets arising from Meyer’s cut and project method [BLM07, Mey72, Moo00, Sch00]. In both cases, the factor map is almost surely injective, so that the dynamics are diam-mean equicontinuous. Examples of mean equicontinuous systems whose factor maps are not almost surely injective are given by certain irregular Toeplitz flows (e.g. [Wil84]) and irregular models sets [FGJO21]. In these cases, the systems are almost automorphic, meaning that the factor maps are almost one-to-one, i.e. the set of injectivity points is residual.

Mean equicontinuous systems for which the factor map to the MEF has no singular fibres are much more difficult to find. In fact, to the best of our knowledge, the only non-trivial examples were so far given by Glasner and Downarowicz in [DG16], who showed that homeomorphisms with these properties are generic in certain spaces of extensions of minimal group rotations. (A ‘trivial’ example would be a homeomorphism of the circle with a unique fixed point. In this case, the MEF is just a single point.) One aim of this note is to provide an alternative construction of such examples based on the well-known Anosov–Katok method [AK70, FK04]. As a byproduct, we also obtain the smoothness of the resulting diffeomorphisms.

Theorem 1.1. There exist $\mathcal {C}^{\infty }$ -diffeomorphisms $\varphi $ of the two-torus with the following properties.

1. (a) $\varphi $ is a skew product over some irrational rotation $R_\alpha :\mathbb {T}^{1}\to \mathbb {T}^{1}, x\mapsto x+\alpha \bmod 1$ .

2. (b) $\varphi $ is totally strictly ergodic (all iterates of $\varphi $ are strictly ergodic) and mean equicontinuous, with the rotation $R_\alpha $ as its maximal equicontinuous factor and the projection to the first coordinate as the factor map.

3. (c) The unique $\varphi $ -invariant measure $\mu $ is the projection of the Lebesgue measure $\unicode{x3bb} $ on $\mathbb {T}^{1}$ onto some measurable graph, that is, it is of the form $\mu =(\mathrm {Id}_{\mathbb {T}^{1}}\times \gamma )_* \unicode{x3bb} $ , where $\gamma :\mathbb {T}^{1}\to \mathbb {T}^{1}$ is measurable.

Note that since the projection to the first coordinate is the factor map to the MEF, all fibres are circles. In particular, there exist no injectivity points. A genericity statement similar to that in [DG16] can also be obtained (see Remark 4.1(a)), but we will not focus on this issue.

The price we have to pay for the smoothness of the examples is that of a more restricted setting. While the construction in [DG16] allows to choose an arbitrary strictly ergodic system as factor, our examples are always extensions of irrational rotations with Liouvillean rotation number. However, on the positive side, the Anosov–Katok construction allows to exert additional control over the lifts of the resulting torus diffeomorphisms to finite covering spaces, and also of all iterates. We can thus ensure that all these mappings are uniquely ergodic. This entails that the finite lifts do not have additional dynamical eigenvalues, so that their discrete spectrum coincides with that of the original system and there has to be a continuous part of the dynamical spectrum. A classical result of Katok and Stepin on cyclic approximations [KS67], combined with further modifications of the construction, allows to ensure that this new part of the spectrum is singular continuous. Altogether, we obtain the following.

Theorem 1.2. For any $m\in \mathbb {N}$ with $m \geqslant 2$ , there exist $\mathcal {C}^{\infty }$ -diffeomorphisms $\varphi $ of the two-torus with the following properties.

1. (a) $\varphi $ is a skew product over some irrational rotation $R_\alpha :\mathbb {T}^{1}\to \mathbb {T}^{1}, x\mapsto x+\alpha \bmod 1$ .

2. (b) $\varphi $ is totally strictly ergodic and a measure-theoretic m to $1$ extension of the irrational rotation $R_\alpha $ , which is the maximal equicontinuous factor of the system.

3. (c) The unique $\varphi $ -invariant measure $\mu $ is the projection of the Lebesgue measure $\unicode{x3bb} $ on $\mathbb {T}^{1}$ onto some m-valued measurable graph, that is, it is of the form $\mu =\sum _{j=1}^m (\mathrm {Id}_{\mathbb {T}^{1}}\times \gamma _j)_* \unicode{x3bb} $ , where $\gamma _j:\mathbb {T}^{1}\to \mathbb {T}^{1}$ are measurable functions for $j=1,\ldots , m$ and $\gamma _i(x)\neq \gamma _j(x)\ \unicode{x3bb} $ -almost surely for all $i\neq j$ . (We call such systems $m:1$ -topomorphic extensions (of the MEF) in analogy to the notion of topo-isomorphic extensions, for which the topological factor map to the MEF is measure-theoretically one-to-one and hence measure-theoretic isomorphism.)

4. (d) The dynamical spectrum of $\varphi $ is given by the (discrete) dynamical spectrum of $R_\alpha $ and a singular continuous component.

In the case $m=2$ , these examples are measure-theoretically similar to (generalized) Thue–Morse subshifts [Kea68], and also to strictly ergodic irregular Toeplitz flows constructed by Iwanik and Lacroix in [IL94]. In both cases, the systems are also measure-theoretically finite-to-one extensions of their MEF, exhibit the same discrete spectrum as the MEF and equally show an additional singular continuous part of the spectrum. However, the topological structure of these examples is quite different, since Toeplitz flows always have a residual set of injectivity points for the projection to the MEF, whereas almost all fibres over the MEF of the generalized Thue–Morse subshift contain exactly two points.

Note that as measure theoretically finite-to-one extensions of equicontinuous systems, the examples provided by Theorems 1.1 and 1.2 have zero entropy. An interesting question, which we have to leave open here, is to understand if their complexity can be adequately described by some notion of slow entropy on a suitable scale or other topological invariants (see [FGJ16, KT97, Pet16]).

1.1. Structure of the article

In §2, we provide all the required preliminaries on topological dynamics, spectral theory, mean equicontinuity and the Anosov–Katok method. In §3, we show that mean equicontinuity is a $G_\delta $ -property. This observation has been made already in [DG16], but to simplify the Anosov–Katok construction carried out in §4, we provide a different $G_\delta $ -characterization of mean equicontinuity here. Finally, in §5, we discuss how to modify the construction to ensure that all iterates of all finite lifts will still be strictly ergodic, no new dynamical eigenvalues occur and the additional spectral component is singular continuous.

2. Preliminaries

2.1. Topological and measure-preserving dynamics

We refer to standard textbooks such as [Aus88, BS02, KH97, Wal82] for the following basic facts on topological dynamics and ergodic theory. Throughout this article, a topological dynamical system (tds) is a pair $(X,\varphi )$ , where X is a compact metric space and $\varphi $ is a homeomorphism of X. We say $\varphi $ (or $(X,\varphi )$ ) is minimal if there exists no non-empty $\varphi $ -invariant compact subset of X. Equivalently, $\varphi $ is minimal if for all $x\in X$ , the $\varphi $ -orbit $\mathcal {O}_\varphi (x)=\{\varphi ^n(x)\mid n\in \mathbb {Z}\}$ of x is dense in X. The tds $(X,\varphi )$ is called equicontinuous if for any $\varepsilon> 0$ , there exists $\delta> 0$ such that $d_X(x,y) < \delta $ implies $d_X(\varphi ^n(x),\varphi ^n(y))<\varepsilon $ for all $n \in \mathbb {N}$ . In this case, there is an equivalent metric on X such that $\varphi $ becomes an isometry. When $(X,\varphi )$ is both equicontinuous and minimal, then X can be given the structure of a compact abelian group with group operation $\oplus $ such that $\varphi $ is just the rotation by some element from X, that is, there exists $\alpha \in X$ such that $\varphi (x)=x\oplus \alpha $ for all $x\in X$ . We write $x\ominus y$ for $x\oplus (\ominus y)$ in this situation, where $\ominus y$ is the inverse of y. Since minimal rotations on compact abelian groups are always uniquely ergodic, the same holds for minimal equicontinuous systems.

Another tds $(Y,\psi )$ is called a factor of $(X,\varphi )$ with factor map $\pi :X\to Y$ if $\pi $ is continuous and onto and satisfies $\pi \circ \varphi =\psi \circ \pi $ . If in addition $\pi $ is a homeomorphism, we say $(X,\varphi )$ and $(Y,\psi )$ are conjugate. Note that both minimality and equicontinuity are inherited by factors. Since factor maps are in general not unique, we will sometimes also refer to the triple $(Y,\psi ,\pi )$ as a factor to specify the factor map. We call such a triple a maximal equicontinuous factor (MEF) of $(X,\varphi )$ if $(Y,\psi )$ is equicontinuous and for any other equicontinuous factor $(Z,\rho ,p)$ , there exists a unique factor map q between $(Y,\psi )$ and $(Z,\rho )$ such that $p=q\circ \pi $ . The existence of an MEF is ensured by the following statement, which also addresses the question of uniqueness.

Theorem 2.1. [Aus88, Theorem 9.1, p. 125]

Every topological dynamical system $(X,\varphi )$ has an MEF $(Y,\psi ,\pi )$ . If $(\hat Y, \hat \psi ,\hat \pi )$ is another MEF, then there is a unique conjugacy $h: (Y,\psi ) \to (\hat Y,\hat \psi )$ such that $\hat \pi =h \circ \pi $ . In particular, the systems $(Y,\psi )$ and $(\hat Y,\hat \psi )$ are conjugate in this case.

Remark 2.2.

1. (a) Despite the lack of uniqueness, we will often refer to an MEF $(Y,\psi )$ of $(X,\varphi )$ as ‘the MEF’, in particular, in situations where we are only interested in conjugacy-invariant properties.

2. (b) Note also that once we have fixed an equicontinuous tds $(Y,\psi )$ as the MEF of a minimal system $(X,\varphi )$ , the corresponding factor map $\pi $ is unique modulo post-composition with a rotation on Y (where we refer to the above-mentioned group structure of minimal equicontinuous systems). The reason is the fact that, in this case, given two different factor maps $\pi _1,\pi _2:X\to Y$ , the Y-valued function $\pi _1\ominus \pi _2$ is continuous and $\varphi $ -invariant, and therefore constant by minimality.

A measure-preserving dynamical system (mpds) is a quadruple $(X,\mathcal {A},\mu ,\varphi )$ consisting of a probability space $(X,\mathcal {A},\mu )$ and a measurable transformation $\varphi :X\to X$ that preserves the measure $\mu $ , that is, $\varphi _*\mu =\mu $ , where $\varphi _*\mu (A)=\mu (\varphi ^{-1}(A))$ . An mpds is ergodic if every $\varphi $ -invariant set $A\in \mathcal {A}$ has measure $0$ or $1$ . This is equivalent to the validity of the assertion of the Birkhoff ergodic theorem: for any $f\in L^1(\mu )$ , there holds

(2.1) $$ \begin{align} \lim_{n\rightarrow\infty} \frac{1}{n} \sum_{i=0}^{n-1} f\circ \varphi^i(x) = \int_X f\,d\mu \end{align} $$

for $\mu $ -almost every $x\in X$ . Given two mpds $(X,\mathcal {A},\mu ,\varphi )$ and $(Y,\mathcal {B},\nu ,\psi )$ , we call a measurable map $h:X\to Y$ a measure-theoretic isomorphism if there exist sets $A\in \mathcal {A},\ B\in \mathcal {B}$ such that $\mu (A)=\nu (B)=1$ , $h:A\to B$ is a bi-measurable bijection, $h_*\mu =\nu $ and $\varphi \circ h=\psi \circ h$ on A.

The mpds we consider will mostly be topological, that is, X will be a compact metric space, $\mathcal {A}=\mathcal {B}(X)$ the Borel $\sigma $ -algebra on X, $\mu $ a Borel measure and $\varphi $ a homeomorphism. In particular, this means that $\varphi $ is a bi-measurable bijection. For any tds $(X,\varphi )$ , the existence of at least one $\varphi $ -invariant probability measure is ensured by the Krylov–Bogolyubov theorem. If there exists exactly one invariant measure – which is necessarily ergodic in this case – we call a tds uniquely ergodic. In this case, the uniform ergodic theorem states that the convergence of the ergodic averages in equation (2.1) is uniform for any continuous function f on X. Actually, the same holds if $\varphi $ admits multiple invariant measures, but the integral of the function f is the same with respect to all of them. This is a more or less direct consequence of the Krylov–Bogolyubov procedure and can be extended, to families of continuous functions that are compact in the uniform topology, in the following way.

Theorem 2.3. (Simultaneous uniform ergodic theorem)

Suppose that $(X,\varphi )$ is uniquely ergodic with invariant measure $\mu $ . For any compact family $\mathcal {F} \subseteq \mathcal {C}(X,[0,1])$ of continuous functions and $x \in X$ , the simultaneous ergodic averages

$$ \begin{align*} A_n: X \times \mathcal{F} \longrightarrow \mathbb{R},\quad (x,f) \longmapsto\frac{1}{n}\sum_{i=0}^{n-1} f( \varphi^i(\cdot) ) \end{align*} $$

converge uniformly to the function $(x,f) \mapsto \int f\,d\mu $ .

We omit the proof, which is a straightforward adaptation of the standard argument for the uniform ergodic theorem (see e.g. [Wal82, Theorem 6.19]).

2.2. Spectral theory of dynamical systems

Given an mpds $(X,\mathcal {A},\mu ,\varphi )$ , the associated Koopman operator is given by

$$ \begin{align*} U_{\varphi} : L^2_\mu(X)\to L^2_\mu(X),\quad f\mapsto f\circ\varphi. \end{align*} $$

Since $\mu $ is $\varphi $ -invariant, $U_\varphi $ is a unitary operator, so that $\sigma (U_\varphi )\subseteq \mathbb {S}^1$ . It is well known that spectral properties of $U_\varphi $ are closely related to dynamical properties of the system. For instance, ergodicity of $\varphi $ is equivalent to the simplicity of $1$ as an eigenvalue, and weak mixing of the system is equivalent to the absence of any further eigenvalues.

For any continuous function $f:\sigma (U_\varphi )\to \mathbb {C}$ , the continuous functional calculus yields the existence of a bounded linear operator $f(U_\varphi )$ on $L^2_\mu (X)$ such that the mapping

$$ \begin{align*} \mathcal{C}(\sigma(U_\varphi),\mathbb{C})\to \mathcal{C}^*(U_\varphi) = \{f(U_\varphi)\mid f\in\mathcal{C}(\sigma(U_\varphi),\mathbb{C})\},\quad f\mapsto f(U_\varphi) \end{align*} $$

is an isomorphism of $\mathcal {C}^*$ -algebras. Given $g\in L^2_\mu (X)$ , this further allows to define a bounded linear functional

$$ \begin{align*} \ell_g : \mathcal{C}(\sigma(U_\varphi),\quad\mathbb{C})\to \mathbb{C},\quad f\mapsto \langle f(U_\varphi)g,g\rangle \end{align*} $$

and thus, by virtue of the Riesz representation theorem, a Borel measure $\mu _g$ on $\sigma (U_\varphi )\subseteq~\mathbb {S}^1$ such that

$$ \begin{align*} \langle f(U_\varphi)g,g\rangle = \int_{\sigma(U_\varphi)} f\,d\mu_g. \end{align*} $$

The measure $\mu _g$ is called the spectral measure associated to g. Moreover, there exists an orthogonal decomposition

$$ \begin{align*} L^2_\mu(X) = L^2_\mu(X)_{\mathrm{pp}} \oplus L^2_\mu(X)_{\mathrm{sc}} \oplus L^2_\mu(X)_{\mathrm{ac}} \end{align*} $$

of $L^2_\mu (X)$ , where

$$ \begin{align*} L^2_\mu(X)_{\mathrm{pp}} & = \{ g\in L^2_\mu(X) \mid \mu_g \textrm{ is a pure point}\} , \\ L^2_\mu(X)_{\mathrm{sc}} & = \{ g\in L^2_\mu(X) \mid \mu_g \textrm{ is singular continuous}\} , \\ L^2_\mu(X)_{\mathrm{ac}} & = \{ g\in L^2_\mu(X) \mid \mu_g \textrm{ is absolutely continuous}\}. \end{align*} $$

In our setting, the reference measure with respect to which the singularity and absoluteness (of continuity) is defined is the Lebesgue measure on the circle.

The spectra $\sigma _{\mathrm {pp}}(U_\varphi )$ , $\sigma _{\mathrm {sc}}(U_\varphi )$ and $\sigma _{\mathrm {ac}}(U_\varphi )$ obtained from the restriction of $U_\varphi $ to these subspaces are called the discrete/pure point, singular continuous and absolutely continuous part, or component, of the dynamical spectrum of $U_\varphi $ . Note that the different spectral parts need not be disjoint.

In the case of a purely discrete spectrum, it turns out that a system is uniquely characterized, up to isomorphism, by the group of its dynamical eigenvalues.

Theorem 2.4. (Halmos and von Neumann [HvN42, vN32])

An ergodic mpds $(X,\mathcal {A},\mu ,\varphi )$ has purely discrete spectrum if and only if it is measure-theoretically isomorphic to a minimal rotation of a compact abelian group equipped with its Haar measure. Moreover, two mpds with purely discrete spectrum are isomorphic if and only if they have the same group of eigenvalues.

To prove the existence of a singular continuous spectral component, we will use a classical result from the theory of approximations by periodic transformations presented in [KS67]. An mpds $(X,\mathcal {A},\mu ,\varphi )$ admits cyclic approximation by periodic transformations (capt) with speed $s:\mathbb {N}\to \mathbb {R}^+_0$ if there exists a sequence $(\varphi _n)_{n\in \mathbb {N}}$ of bijective bi-measurable transformations on X and a sequence $(\mathcal {P}_n)_{n\in \mathbb {N}}$ of finite measurable partitions of X, $\mathcal {P}_n=\{P_{n,1},\ldots , P_{n,K_n}\}$ , such that for all $n\in \mathbb {N}$ :

1. (P1) $\varphi _n$ cyclically permutes the elements of $\mathcal {P}_n$ ;

2. (P2) for each $A\in \mathcal {A}$ , there exist $A_n\in \sigma (\mathcal {P}_n)$ such that $\mu (A\Delta A_n)\stackrel {n\rightarrow \infty }{\longrightarrow } 0$ ;

3. (P3) $\sum _{i=1}^{K_n} \mu (\varphi (P_{n,i})\Delta \varphi _n(P_{n,i})) \leq s(K_n)$ .

Here, $\sigma (\mathcal {P}_n)$ denotes the $\sigma $ -algebra generated by $\mathcal {P}_n$ .

Theorem 2.5. [KS67, Corollary 3.1]

If an mpds $(X,\mathcal {A},\mu ,\varphi )$ admits capt with speed $s:\mathbb {N}\to \mathbb {R}^+_0$ and $\lim _{n\rightarrow \infty } ns(n)=0$ , then $U_\varphi $ has no absolutely continuous spectral component, that is, $\sigma _{\mathrm {ac}}(U_\varphi )=\emptyset $ .

2.3. Mean equicontinuity

A tds $(X,\varphi )$ is called mean equicontinuous if for all $\varepsilon> 0$ , there is $\delta> 0$ such that $d_X(x,y)<\delta $ implies

(2.2) $$ \begin{align} d_{\mathsf{B}}(x,y) = \varlimsup_{n\to\infty} \frac{1}{n} \sum_{i=0}^{n-1} d(\varphi^i(x),\varphi^i(y)) < \varepsilon. \end{align} $$

It turns out that in the minimal case, mean equicontinuity implies unique ergodicity and is moreover equivalent to a certain invertibility property of the factor map onto the MEF.

Theorem 2.6. [DG16, Theorem 2.1]

Suppose $(X,\varphi )$ is minimal and $(Y,\psi ,\pi )$ is an MEF. Denote by $\nu $ the unique invariant measure of $(Y,\psi )$ . Then the following are equivalent.

1. (1) $(X,\varphi )$ is mean equicontinuous.

2. (2) $(X,\psi )$ is uniquely ergodic with unique invariant measure $\mu $ and $\pi $ is a measure-theoretic isomorphism between the mpds $(X,\mathcal {B}(X),\mu ,\varphi )$ and $(Y,\mathcal {B}(Y), \nu , \psi )$ .

Given a uniquely ergodic tds $(X,\varphi )$ and a factor $(Y,\psi ,\pi )$ (which is automatically uniquely ergodic as well), we say $(X,\varphi )$ is an isomorphic extension of $(Y,\psi )$ if $\pi $ is a measure-theoretic isomorphism between the two mpds $(X,\mathcal {B}(X),\mu ,\varphi )$ and $(Y,\mathcal {B}(Y),\nu ,\psi )$ , where $\mu $ and $\nu $ are the unique invariant measures for $\varphi $ and $\psi $ , respectively. Hence, condition (2) above can be rephrased by saying that $(X,\varphi )$ is an isomorphic extension of $(Y,\psi )$ .

Note that the above statement implies, in particular, that the dynamical spectrum of mean equicontinuous systems coincides with that of the MEF and is therefore purely discrete.

The mapping $d_{\mathsf {B}}:X\times X\to \mathbb {R}^+_0$ defined in equation (2.2) is always a pseudo-metric on X. It is called the Besicovitch pseudo-metric. For mean equicontinuous systems, it provides a way to directly define an MEF of the system.

Proposition 2.7. [DG16]

Suppose $(X,\varphi )$ is mean equicontinuous. Define an equivalence relation on X by

$$ \begin{align*} x \sim y \Leftrightarrow d_{\mathsf{B}}(x,y) = 0. \end{align*} $$

Then the quotient system $(X/\sim ,\varphi /\sim )$ together with the canonical projection as a factor map is an MEF.

The proof of this fact in [DG16] is implicit – it is contained in the proof of [DG16, Theorem 2.1].

2.4. The Anosov–Katok method

The Anosov–Katok method is arguably one of the best-known and most widely used constructions in smooth dynamics and allows to obtain a broad scope of examples with particular combinations of dynamical properties. Although many readers will already be familiar with the general method, we provide a brief introduction to fix notation and comment on some specific issues that will be relevant in our context. The construction of mean equicontinuous diffeomorphism of the two-torus will then be carried out in §4, while the modification required to obtain the finite-to-one extensions in Theorem 1.2 will be discussed in §5.

We restrict to the case of tori $\mathbb {T}^d=\mathbb {R}^d/\mathbb {Z}^d$ and denote by $\mathrm {Homeo}(\mathbb {T}^d)$ the space of homeomorphisms of the d-dimensional torus, by $\mathcal {C}^k(\mathbb {T}^d)$ the space of k-times differentiable torus endomorphisms (including the cases $k=\infty $ and $k=\omega $ , were the latter stands for ‘real-analytic’) and let

$$ \begin{align*} \mathrm{Diffeo}^k(\mathbb{T}^d) = \{ \varphi\in\mathrm{Homeo}(\mathbb{T}^d) \mid \varphi,\varphi^{-1}\in\mathcal{C}^k(\mathbb{T}^d)\}. \end{align*} $$

We will identify $\mathrm {Diffeo}^0(\mathbb {T}^d)$ and $\mathrm {Homeo}(\mathbb {T}^d)$ . Further, we denote the supremum metric on $\mathcal {C}^0(\mathbb {T}^d)$ by $d_{\sup }$ and let

$$ \begin{align*} d_k(\varphi,\psi) = \max_{i=0}^k\max \{d_{\sup}(\varphi^{(i)},\psi^{(i)}), d_{\sup}((\varphi^{-1})^{(i)}, (\psi^{-1})^{(i)})\} \end{align*} $$

be the standard metric on the space of torus diffeomorphisms. By $B^k_\varepsilon (\psi )$ , we denote the $\varepsilon $ -ball around $\psi $ in $\mathrm {Diffeo}^k(\mathbb {T}^d)$ .

Our aim is to recursively construct a sequence $(\varphi _n)_{n\in \mathbb {N}}$ of torus diffeomorphisms according to the following scheme.

• • Each $\varphi _n$ will be of the form $H_n\circ R_{\rho _n}\circ H_n^{-1}$ , where $R_{\rho } :\mathbb {T}^d\to \mathbb {T}^d,\ x\mapsto x+\rho $ denotes the rotation with rotation vector $\rho \in \mathbb {T}^d$ and $H_n\in \mathrm {Diffeo}^{\infty }( \mathbb {T}^d)$ .

• • The $H_n$ will be of the form $H_n=h_1\circ \cdots \circ h_n$ , where each $h_n\in \mathrm {Diffeo}^{\infty }(\mathbb {T}^d)$ commutes with the rotation $R_{\rho _{n-1}}$ , that is, $h_n\circ R_{\rho _{n-1}} = R_{\rho _{n-1}} \circ h_n$ . Note that consequently, we have that $H_n\circ R_{\rho _{n-1}}\circ H_n^{-1}=\varphi _{n-1}$ . Hence, at this stage, we have introduced the new conjugating map $h_n$ , but the system has not changed yet.

• • When going from step $n-1$ to n in the construction, we choose $h_n$ first and only pick the new rotation vector $\rho _n$ afterwards. Therefore, the continuity of the mapping $\rho \mapsto H_n\circ R_\rho \circ H_{n}^{-1}$ (with respect to the metric $d_k$ for any $k\in \mathbb {N}$ ) allows to control the difference between $\varphi _{n-1}$ and $\varphi _n$ in the respective metric.

• • As a consequence, we can ensure that the resulting sequence $(\varphi _n)_{n\in \mathbb {N}}$ is Cauchy in $\mathrm {Diffeo}^k(\mathbb {T}^d)$ for any $k\in \mathbb {N}$ , simply by recursively choosing $\rho _n$ sufficiently close to $\rho _{n-1}$ with respect to $d_n$ in the nth step of the construction. This ensures that the $\varphi _n$ converge to some limit $\varphi \in \mathrm {Diffeo}^{\infty }(\mathbb {T}^d)$ .

So far, the above items explain how to ensure the convergence of the constructed sequence $(\varphi _n)_{n\in \mathbb {N}}$ , but they do not yet specify how to obtain any particular dynamical properties. It turns out, however, that the method is tailor-made to realize any $G_\delta $ -properties in the space $\mathrm {Homeo}(\mathbb {T}^d)$ . Suppose we want to ensure that our limit diffeomorphism $\varphi $ belongs to a set of torus homeomorphism $A\subseteq \mathrm {Homeo}(\mathbb {T}^d)$ which is $G_\delta $ , that is, it is of the form $A=\bigcap _{n\in \mathbb {N}} U_n$ with $U_n\subseteq \mathrm {Homeo}(\mathbb {T}^d)$ open. As discussed below, both minimal and uniquely ergodic torus homeomorphisms can be characterized in this way. We then proceed as follows.

1. (i) By choosing $h_n$ and $\rho _n$ accordingly, we ensure that $\varphi _n\in U_n$ . How this is done exactly depends on the property that defines A. This is actually the crucial step in the construction and we will provide details further below.

2. (ii) Since $U_n$ is open, there exists $\eta _n>0$ such that $\overline {B^0_{\eta _n}(\varphi _n)}\subseteq U_n$ . By ensuring that the distance between $\varphi _j$ and $\varphi _{j+1}$ is small and decays sufficiently fast for all $j\geq n$ , this yields $\varphi =\lim _{j\to \infty } \varphi _j\in \overline {B^0_{\eta _n}(\varphi _n)}\subseteq U_n$ as well. Since this works for all $n\in \mathbb {N}$ , we obtain $\varphi \in A$ .

3. (iii) To ensure $\varphi _n\in U_n$ , it will often be convenient not to go from $\varphi _{n-1}$ to $\varphi _n$ directly, but to pass through some intermediate map $\hat \varphi _{n-1}$ instead. This is, for example, useful to ensure that the limit system is minimal and/or uniquely ergodic. For instance, we may first choose a totally irrational rotation number $\hat \rho _{n-1}$ and define $\hat \varphi _{n-1}=H_n\circ R_{\hat \rho _{n-1}}\circ H_n^{-1}$ . Then, $\hat \varphi _{n-1}$ is minimal and uniquely ergodic, since this is true for the irrational rotation $R_{\hat \rho _{n-1}}$ . If $U_n$ is defined in such a way that it contains all minimal/uniquely ergodic torus homeomorphisms, then $\hat \varphi _{n-1}\in U_n$ is automatic. If $\varphi _n$ is then chosen sufficiently close to $\hat \varphi _{n-1}$ (by choosing $\rho _n$ close to $\hat \rho _{n-1}$ ), we obtain $\varphi _n\in U_n$ , as required. Further, if both $\hat \rho _{n-1}$ and $\rho _n$ are close enough to $\rho _{n-1}$ , then $\varphi _n$ will also be close to $\varphi _{n-1}$ in $\mathrm {Diffeo}^k(\mathbb {T}^d)$ .

4. (iv) In §4, we will actually use a further modification of the above scheme and pass through an additional third map $\tilde \varphi _{n-1}=H_n\circ R_{\tilde \rho _{n-1}}\circ H_n^{-1}$ , where $\tilde \rho _{n-1}$ is irrational, but not totally irrational (so the entries of the rotation vector are rationally related).

The following works in high generality for any compact metric space X. Using a very similar argument as made in item (ii), we obtain the following proposition.

Proposition 2.8. Let $A = \bigcap _{n\in \mathbb {N}} U_n \subseteq {\mathrm {Homeo}}(X) =: Z$ , where the $U_n$ are open with respect to $d_0$ . There exists a sequence $(\eta ^A_n)_{n \in \mathbb {N}}$ of functions $\eta _n^A: Z^n \longrightarrow \mathbb {R}^+$ such that if a sequence $(\varphi _n)_{n\in \mathbb {N}} \in Z^{\mathbb {N}}$ satisfies

(2.3) $$ \begin{align} d_0( \varphi_n,\varphi_{n+1}) < {\eta^A_n ( \varphi_1,\ldots,\varphi_n)}\quad\text{and}\quad \varphi_n \in U_n \end{align} $$

for all $n\in \mathbb {N}$ , then it converges with limit $\varphi =\lim _{n\rightarrow \infty } \varphi _n \in A$ .

Sketch of proof

If we set $\eta ^A_n < 2^{-n}$ , any sequence satisfying equation (2.3) will be Cauchy and converge to some $\varphi \in Z$ as Z is complete. The idea is now to choose $\eta ^A$ small enough such that equation (2.3) implies $\varphi _n \in U_k$ for $n\geqslant k$ . This can be done by fixing some $\delta _k> 0$ such that $\overline {B_{\delta _k}(\varphi _k)} \subseteq U_k$ and then making sure that equation (2.3) implies $d_0(\varphi _n,\varphi _k) < \delta _k$ , i.e. $\varphi _n \in B_{\delta _k}(\varphi _k)$ . One way to do that and to encode all the previously imposed conditions into $\eta ^A_n$ is to recursively ensure that

(2.4) $$ \begin{align} \eta^A_n(\varphi_1,\ldots,\varphi_n) < \min \bigg( \frac{\delta_n}{2}, \min_{k=1}^{n-1}( \eta^A_k(\varphi_1,\ldots,\varphi_k) - d_0(\varphi_k,\varphi_n) ) \bigg). \end{align} $$

Now we argue that if $(\varphi _n)_{n\in \mathbb {N}}$ satisfies equation (2.3), then equation (2.4) inductively implies

(2.5) $$ \begin{align} \eta^A_k(\varphi_1,\ldots,\varphi_k)> d_0(\varphi_k,\varphi_n) \end{align} $$

for any $n> k$ , and thus both $\eta ^A_n> 0$ and $\varphi _n \in B_{\delta _k}(\varphi _k)$ (as $\eta ^A_k(\varphi _1,\ldots ,\varphi _k) < \delta _k$ ). For $n=k+1$ , equation (2.5) holds trivially. Now for the induction step $n \to n+1$ , we have that

$$ \begin{align*} d_0( \varphi_{n+1},\varphi_{n}) < \eta^A_n ( \varphi_1,\ldots,\varphi_n ) < \eta^A_k(\varphi_1,\ldots,\varphi_k) - d_0(\varphi_k,\varphi_n). \end{align*} $$

The triangle inequality now implies equation (2.5).

In total, $\varphi _m \in B_\eta (\varphi _n) \subseteq \overline {B_\eta (\varphi _n)}$ for all $m> n$ . So $\varphi \in \overline {B_\eta (\varphi _n)} \subseteq U_n$ for all $n \in \mathbb {N}$ and thus $\varphi \in \bigcap _{n\in \mathbb {N}}U_n = A$ .

2.5. $G_\delta $ -characterization of strict ergodicity

It is well known that the set $\mathrm {Homeo}^{\mathrm {se}}(X)$ of strictly ergodic homeomorphisms of X is $G_\delta $ in $\mathrm {Homeo}(X)$ . For the convenience of the reader, we include a short proof of this folklore result.

We first show that minimality is $G_\delta $ . Let $\varepsilon> 0$ . A set $A \subseteq X$ is called $\varepsilon $ -dense if $B_\varepsilon (A) = X$ . Observe that the mapping $\xi $ is minimal if and only if for any $\varepsilon> 0$ , there is $M \in \mathbb {N}$ such that $\{ x, \varphi (x), \ldots , \varphi ^{M}(x) \}$ is $\varepsilon $ -dense for any $x \in X$ . Furthermore, the kth iterate of a homeomorphism depends continuously on that homeomorphism. Therefore,

$$ \begin{align*} U^{\min}_{M,\varepsilon} = \{ \psi \in {\mathrm{Homeo}}(X,X)\mid\text{for all } x \in X: \{ x, \psi(x), \ldots, \psi^M(x) \} \text{ is }\varepsilon\text{-dense} \} \end{align*} $$

is open in the supremum norm. If $\mathrm {Homeo}^{\min }(X)$ denotes the set of all minimal homeomorphisms of X, the above yields

$$ \begin{align*} \mathrm{Homeo}^{\min}(X) = \bigcap_{\varepsilon \in \mathbb{Q}^+} \bigcup_{M\in\mathbb{N}} U^{\min}_{M,\varepsilon} \,. \end{align*} $$

This means in particular that $\mathrm {Homeo}^{\min }(X)$ is $G_\delta $ .

Now, we turn to unique ergodicity. Fix a dense set $\{ s_n \mid n \in \mathbb {N}\} \subset \mathcal {C}(X,\mathbb {R})$ . Given $n \in \mathbb {N}$ and a continuous map $\xi :X\to X$ , we can assign to any $g \in \mathcal {C}(X,\mathbb {R})$ its n-step ergodic average

$$ \begin{align*} A_n^\xi g = \frac{1}{n}\sum_{i=0}^{n-1} g\circ \xi^i. \end{align*} $$

It is well known (e.g. [EW10, Theorem 4.10, p. 105]) that $\xi $ is uniquely ergodic if and only if for every $k \in \mathbb {N}$ , the sequence of functions $(A_n^\xi s_k)_{n\in \mathbb {N}}$ converges pointwise to a constant. For $g \in \mathcal {C}(X,\mathbb {R})$ , we denote by $V(g)$ its variation over X, that is,

$$ \begin{align*} V(g) := \sup_{x\in X} g(x)- \inf_{y \in X} g(y). \end{align*} $$

Given any $\psi $ -invariant probability measure $\mu $ , we have $\int A_n^\psi g\,d\mu = \int g\,d\mu $ . This implies that if $V(A_n^\psi g) \xrightarrow {n\to \infty } 0$ , then $A_n^\psi g \xrightarrow {n\to \infty } \int g\,d\mu $ .

As $A_k^\xi g$ depends continuously on $\xi $ and $V(f)$ depends continuously on f, we see that

$$ \begin{align*} U^{\mathrm{ue}}_{n,K,\beta} := \{ {\psi} \in {\mathrm{Homeo}}(X,X) \mid V(A_K^{\psi} s_n) < \beta \} \end{align*} $$

is open in the supremum metric. In particular, the set of those transformations for which the ergodic average of $s_n$ eventually stabilizes at a variation below $\beta $ , given by

$$ \begin{align*} U^{\mathrm{ue}}_{n,\beta} = \bigcup_{K \in \mathbb{N}} \bigcap_{k> K} U^{\mathrm{ue}}_{n,k,\beta}, \end{align*} $$

is $G_\delta $ . We notice that $\psi \in \bigcap _{k\in \mathbb {N}} \bigcap _{\beta \in \mathbb {Q}^+} U^{\mathrm {ue}}_{k,\beta }$ if and only if $A_n^\psi s_n$ converges to $\int s_n\,d\mu $ for any $n \in \mathbb {N}$ if and only if $\psi $ is uniquely ergodic. This yields that the set $\mathrm {Homeo}^{\mathrm {ue}}(X)$ of uniquely ergodic homeomorphisms of X is $G_\delta $ .

Altogether, this shows that $\mathrm {Homeo}^{\mathrm {se}}(X)=\mathrm {Homeo}^{\min (X)}\cap \mathrm {Homeo}^{\mathrm {ue}}(X)$ is a $G_\delta $ -set as claimed.

3. $G_\delta $ -characterization of mean equicontinuity for skew products

As discussed in the previous section, to construct mean equicontinuous systems via the Anosov–Katok method, it is instrumental to have an explicit $G_\delta $ -characterization of mean equicontinuity available. In principle, such a characterization is already contained in [DG16]. However, the latter uses the fact that mean equicontinuity is equivalent to the existence of a unique self-joining on the product space $X\times X$ over the MEF as a common factor. Since we want to avoid working in the product space, as this would rather complicate the construction in the next section, we provide an alternative characterization here. As in [DG16], we make use of the fact that we are in a skew product setting and the factor map is given a priori (by the projection to the first coordinate). We formulate the statement in abstract terms, as it might be useful in other situations as well.

Proposition 3.1. Let $(X,\varphi )$ be a tds and $(Y,\psi ,\pi )$ an equicontinuous factor. Then $(Y,\psi ,\pi )$ is an MEF of $(X,\varphi )$ and $(X,\varphi )$ is an isomorphic extension of $(Y,\psi )$ if and only if for all $\varepsilon> 0$ , there exists some $K \in \mathbb {N}$ such that, for all $x,y\in X$ , we have

(3.1) $$ \begin{align} \pi(x)=\pi(y) \Longrightarrow \frac 1{K} \sum_{i=0}^{K} d_X(\varphi^i(x),\varphi^i(y)) < \varepsilon. \end{align} $$

Remark 3.2. (a) Denote by $\mathrm {Homeo}^{\mathrm {eq}}(Y)$ the space of equicontinuous homeomorphisms of Y. Consider the space

$$ \begin{align*} \mathcal{E}(\pi) = \{\varphi\in\mathrm{Homeo}(X)\mid \text{there exists } \psi\in\mathrm{Homeo}^{\mathrm{eq}}(Y): \pi\circ\varphi=\psi\circ \pi\} \end{align*} $$

with the subspace

$$ \begin{align*} \mathcal{E}^{\mathrm{iso}}(\pi) = \bigg\{\varphi\in\mathcal{E}(\pi)\ \bigg| \begin{array}{c@{}}\displaystyle \text{there exists } \psi\in\mathrm{Homeo}(Y): (Y,\psi,\pi) \textrm{ is the MEF of } (X,\varphi)\\ \quad \displaystyle\textrm{and } (X,\varphi) \textrm{ is its isomorphic extension} \end{array} \bigg\}. \end{align*} $$

Let

$$ \begin{align*} U_n^{\mathrm{iso}}(\pi) = \{ \varphi\in\mathcal{E}(\pi)\mid \text{there exists } K\in\mathbb{N}: \text{ equation}~(3.1) \textrm{ holds with } \varepsilon=1/n \}. \end{align*} $$

Then, by the above statement, we have $\mathcal {E}^{\mathrm {iso}}(\pi )=\bigcap _{n\in \mathbb {N}} U_n^{\mathrm {iso}}(\pi )$ . As the sets $U_n^{\mathrm {iso}}(\pi )$ are open, this implies that $\mathcal {E}^{\mathrm {iso}}(\pi )$ is a $G_\delta $ -set in $\mathcal {E}(\pi )$ .

Note here that the property in equation (3.1) only depends on the factor map $\pi $ , but not on the map $\psi $ acting on the factor space.

(b) A similar characterization could be given with a fixed factor system $(Y,\psi )$ on the base. However, in the context of the Anosov–Katok construction, where the base systems of the approximating diffeomorphisms will be circle rotations with varying rotation numbers, the independence of $\psi $ in the above characterization is crucial.

(c) According to Proposition 2.8, there are mappings

$$ \begin{align*} \eta_n^{\mathrm{me}} : {\mathrm{Homeo}}(\mathbb{T}^d)^n \longrightarrow [0,2^{-n}] \end{align*} $$

such that if a sequence $(\tilde \varphi _n)_{n\in \mathbb {N}}$ in $\mathcal {E}(\pi )$ satisfies $\tilde \varphi _n\in U_n^{\mathrm {iso}}(\pi )$ and

$$ \begin{align*} d_0( \tilde\varphi_n,\tilde\varphi_{n+1}) < {\eta^{\mathrm{me}}_n( \tilde\varphi_1,\ldots,\tilde\varphi_n )} \end{align*} $$

for all $n\in \mathbb {N}$ , then its limit $\varphi =\lim _{n\rightarrow \infty } \tilde \varphi _n$ exists and belongs to $\mathcal {E}^{\mathrm {iso}}(\pi )$ .

Proof of Proposition 3.1

First, assume that $( Y,\psi ,\pi )$ is an MEF and $( X,\varphi )$ is an isomorphic extension of $( Y,\psi )$ . Then, by Theorem 2.7, $\pi (x)=\pi (y)$ implies $d_{\mathsf {B}}(x,y)=0$ . Let

$$ \begin{align*} \mathcal{J}_\pi( X) = \{(x,y)\in X\times X\mid \pi(x)=\pi(y)\}. \end{align*} $$

Then $d_{\mathsf {B}}(x,y)$ is the ergodic average of the function $d_X$ for the action of $\varphi \times \varphi $ on $X\times X$ . Since these ergodic averages are identically zero on the compact invariant set $\mathcal {J}_\pi (X)$ , we have that $\int _{\mathcal {J}_\pi (X)}\,d_X(x,y)\,d\gamma (x,y)=0$ for all $\varphi \times \varphi $ -invariant measures $\gamma $ on $\mathcal {J}_\pi (X)$ (in fact, by [DG16, Proposition 2.5], there is only one such measure when $(X,\varphi )$ is mean equicontinuous). The uniform ergodic theorem therefore implies that the functions

$$ \begin{align*} a_K(x,y) = \frac 1{K} \sum_{i=0}^{K}\,d_X(\varphi^i(x),\varphi^i(y)) \end{align*} $$

uniformly converge to zero as $K\to \infty $ . Hence, we have $a_K<\varepsilon $ for sufficiently large $K\in \mathbb {N}$ , which is just an equivalent reformulation of equation (3.1).

Conversely, suppose that for all $\varepsilon>0$ , there exists $K\in \mathbb {N}$ such that equation (3.1) holds. We assume without loss of generality that $\psi $ is an isometry. Denote by $\Delta _Y$ and $\Delta _X$ the diagonals in the respective product spaces $Y\times Y$ and $X\times X$ . Note that thus $\mathcal {J}_\pi (X)= (\pi \times \pi )^{-1}(\Delta _Y)$ .

Now, fix $\varepsilon>0$ and choose $K\in \mathbb {N}$ according to equation (3.1). This means that the function $a_K$ is strictly smaller than $\varepsilon $ on $\mathcal {J}_\pi (X)$ . By compactness of $\mathcal {J}_\pi (X)$ and continuity of $a_K$ , there exists $\delta _1>0$ such that $a_K<\varepsilon $ on $B_{\delta _1} \ ( J_\pi (X))$ . Due to the continuity of $\pi \times \pi $ , there exists $\eta>0$ such that $A = (\pi \times \pi )^{-1}(\overline {B_\eta (\Delta _Y)})\subseteq B_{\delta _1}(\mathcal {J}_\pi (X))$ . Further, as $\pi $ is uniformly continuous, there exists $\delta>0$ such that $\pi \times \pi ( B_\delta ( \mathcal {J}_\pi (X))) \subseteq B_\eta (\Delta _Y)$ .

As $\psi $ is an isometry, the set A is $\varphi \times \varphi $ -invariant. Since $d_{\mathsf {B}}$ is equal to the ergodic average of $a_K$ for the action of $(\varphi \times \varphi )^K$ and $a_K<\varepsilon $ on A, this yields $d_{\mathsf {B}}<\varepsilon $ on A. However, by the above choices, $d_X(x,y)<\delta $ implies $d_Y(\pi (x),\pi (y))<\eta $ and therefore $(x,y)\in A$ . Hence, we obtain that $d_X(x,y)<\delta $ implies $d_{\mathsf {B}}(x,y)<\varepsilon $ . As $\varepsilon>0$ was arbitrary, this means that $(X,\varphi )$ is mean equicontinuous.

Therefore, Theorem 2.6 implies that $(X,\varphi )$ is an isomorphic extension of its MEF, which we denote by $(\hat Y,\hat \psi ,\hat \pi )$ . By definition, we know that $(Y,\psi )$ is a factor of the MEF, so the fibres of $\hat \pi $ are contained in the fibres of $\pi $ . However, since the Besicovitch distance $d_{\mathsf {B}}$ between two points is zero on each fibre of $\pi $ , and as this property characterizes the fibres of the MEF due to Proposition 2.7 (note here that the fibres do not depend on the particular choice of the MEF), the fibres of $\pi $ are also contained in the fibres of $\hat \pi $ . Thus, $\pi $ and $\hat \pi $ have the same fibres, which implies that $(Y,\psi ,\pi )$ is also an MEF and $(X,\varphi )$ is an isomorphic extension of $(Y,\psi )$ .

4. Mean equicontinuous skew products on the torus: Proof of Theorem 1.1

We are going to construct mean equicontinuous diffeomorphisms of the two-torus which have skew product form

(4.1) $$ \begin{align} \varphi:\mathbb{T}^2\to\mathbb{T}^2,\quad (x,y)\mapsto (x+\alpha,\varphi_x(y)) \end{align} $$

and are such that the underlying irrational rotation $R_\alpha :\mathbb {T}^{1}\to \mathbb {T}^{1},\ x\mapsto x+\alpha $ is the MEF and the factor map is given by the projection $\pi :\mathbb {T}^2\to \mathbb {T}^{1},\ (x,y)\mapsto x$ to the first coordinate. To do so, we employ the Anosov–Katok method as described in Section 2.4 and recursively define sequences of skew product diffeomorphism $(\varphi _n)_{n\in \mathbb {N}}, (\tilde \varphi _n)_{n\in \mathbb {N}}$ and $(\hat \varphi _n)_{n\in \mathbb {N}}$ whose common limit $\varphi $ will satisfy the assertions of Theorem 1.1. The general scheme of our inductive construction will be as follows.

• • The mappings $\varphi _n,\tilde \varphi _n$ and $\hat \varphi _n$ will be of the form

  $$ \begin{align*} \varphi_n = H_n\circ R_{\rho_n}\circ H_n^{-1},\quad \tilde\varphi_n=H_{n+1}\circ R_{\tilde\rho_n}\circ H_{n+1}^{-1} \quad\text{and}\quad \hat\varphi_n=H_{n+1}\circ R_{\hat\rho_n}\circ H_{n+1}^{-1} , \end{align*} $$
  where $\rho _n$ is rational, $\tilde \rho _n=\alpha \rho _n$ with and $\hat \rho _n$ is totally irrational. Further, for technical reasons, we require that
  (4.2) $$ \begin{align} \rho_n = \bigg(\frac{p_n}{q_n},\frac{p_n'}{q_n}\bigg)\quad\textrm{with } p_n,p_n',q_n\in\mathbb{N} \textrm{ relatively prime}. \end{align} $$

• • The conjugating diffeomorphisms $H_n$ will be of the form $H_n=h_1\circ \cdots \circ h_n$ , where $h_{n+1}$ always commutes with the rotation $R_{\rho _n}$ . Moreover, all $h_n$ have skew product structure $h_n:(x,y)\mapsto (x,h_{n,x}(y))$ , with fibre maps $h_{n,x}:\mathbb {T}^{1}\to \mathbb {T}^{1}$ .

• • We choose the functions $\eta ^{\mathrm {se}}_n$ and $\eta ^{\mathrm {me}}_n$ and the sets $U_n^{\mathrm {iso}}(\pi )$ according to Proposition 2.8 and Remark 3.2(c).

• • The approximating torus diffeomorphisms $\varphi _n,\tilde \varphi _n,\hat \varphi _n$ will be chosen such that for each $n\geq 2$ , they satisfy

  (4.3) $$ \begin{align} d_n (\varphi_n,\tilde\varphi_{n}) & \leq \tfrac{1}{3} \min \{\eta^{\mathrm{se}}_{n-1} (\hat\varphi_1,\ldots, \hat\varphi_{n-1}),\eta^{\mathrm{me}}_{n-1}(\tilde\varphi_1,\ldots, \tilde\varphi_{n-1})\}, \end{align} $$
  (4.4) $$ \begin{align} d_n (\tilde\varphi_n,\hat\varphi_{n}) & \leq \tfrac{1}{3} \min \{\eta^{\mathrm{se}}_{n-1}(\hat\varphi_1,\ldots, \hat\varphi_{n-1}),\eta^{\mathrm{me}}_{n}(\tilde\varphi_1,\ldots, \tilde\varphi_{n})\}, \end{align} $$
  (4.5) $$ \begin{align} d_n (\hat\varphi_n,\varphi_{n+1}) & \leq \tfrac{1}{3} \min \{\eta^{\mathrm{se}}_{n}(\hat\varphi_1,\ldots, \hat\varphi_{n}),\eta^{\mathrm{me}}_{n} (\tilde\varphi_1,\ldots, \tilde\varphi_{n})\}, \end{align} $$
  (4.6) $$ \begin{align} \tilde\varphi_n & \in U^{\mathrm{iso}}_n(\pi). \end{align} $$
  Note that these conditions together imply that
  (4.7) $$ \begin{align} d_n(\tilde\varphi_n,\tilde\varphi_{n+1}) & \leq \eta_n^{\mathrm{me}}(\tilde\varphi_1,\ldots, \tilde\varphi_n), \end{align} $$
  (4.8) $$ \begin{align} d_n (\hat\varphi_n,\hat\varphi_{n+1}) & \leq \eta_n^{\mathrm{se}}(\hat\varphi_1,\ldots, \hat\varphi_n), \end{align} $$
  for all $n\in \mathbb {N}$ , which together with equation (4.6) means that the conditions of Proposition 2.8 and Remark 3.2(c) are met, where Proposition 2.8 is applied to the sequence $(\hat \varphi _n)_{n\in \mathbb {N}}$ and Remark 3.2(c) is applied to $(\tilde \varphi _n)_{n\in \mathbb {N}}$ . Note here that the diffeomorphisms $\hat \varphi _n$ are all strictly ergodic, since they are conjugate to a totally irrational torus rotation. Consequently, the common limit $\varphi $ of all the sequences, whose existence is also guaranteed by equations (4.3)–(4.5) (note that $\eta ^{\mathrm {me}}_n,\eta ^{\mathrm {se}}_n\leq 2^{-n}$ ), is both strictly ergodic and mean equicontinuous, with $\pi $ as the factor map to the MEF. The MEF is then given by $(\mathbb {T}^{1},R_\alpha ,\pi )$ , where $\alpha =\lim _{n\rightarrow \infty } \pi (\rho _n)$ .

• • Note that the conditions in equations (4.3)–(4.5) can always be ensured by choosing the rotation vectors $\rho _n,\tilde \rho _n,\hat \rho _n$ and $\rho _{n+1}$ sufficiently close to each other. The reason is the fact that we have $\varphi _n=H_{n+1}\circ R_{\rho _n}\circ H_{n+1}^{-1}$ due to the commutativity between $h_{n+1}$ and $R_{\rho _n}$ combined with the continuity of $\rho \mapsto H_{n+1}\circ R_\rho \circ H_{n+1}^{-1}$ . Therefore, the only issue that remains to be addressed is to ensure that the intermediate maps $\tilde \varphi _n$ are indeed contained in $U^{\mathrm {iso}}_n(\pi )$ .

To start the induction, we let $H_1=h_1=\mathrm {Id}_{\mathbb {T}^2}$ and choose $\varphi _1$ to be an arbitrary rational rotation on $\mathbb {T}^2$ whose rotation vector satisfies equation (4.2). Note that the inductive assumptions in equations (4.3)–(4.5) are all still empty at this point.

Now, suppose that $\varphi _1,\ldots , \varphi _N$ , $\tilde \varphi _1,\ldots ,\tilde \varphi _{N-1}$ and $\hat \varphi _1,\ldots ,\hat \varphi _{N-1}$ have been constructed such that equations (4.3)–(4.6) hold for all $n=1,\ldots , N-1$ . We have $\rho _n=(({p_n}/{q_n}),({p_n'}/{q_n}))$ , where $p_n,p_n',q_n\in \mathbb {N}$ are relatively prime. The aim is to choose $h_{n+1}$ and $\tilde \rho _n$ in such a way that $\tilde \varphi _n\in U^{\mathrm {iso}}_n(\pi )$ , that is, $\tilde \varphi _n$ satisfies equation (3.1) with $\varepsilon =1/n$ .

To that end, we note that orbits of $R_{\rho _n}$ move along closed curves of the form

$$ \begin{align*} L(\rho_n,t) = \{(xp_n/q_n,t+xp_n'/q_n)\mid x\in [0,q_n)\} , \end{align*} $$

which are parametrized by the functions

$$ \begin{align*} \ell_{\rho_n,t}:\mathbb{T}^{1}\to L(t,\rho_n),\quad x\mapsto (xp_n,t+xp_n'). \end{align*} $$

We now dwell on this insight a bit further to see how we need to choose $h_{n+1}$ . First, observe that the mapping $\ell _{\rho _n,t}$ conjugates the one-dimensional rotation $r_{1/q_n}$ on $\mathbb {T}^{1}$ and the restriction of $R_{\rho _n}$ to $L(t,\rho _n)$ , that is, $R_{\rho _n}\circ \ell _{\rho _n,t} = \ell _{\rho _n,t}\circ r_{1/q_n}$ . Consequently, the orbits of $\varphi _n=H_{n+1}\circ R_{\rho _n}\circ H_{n+1}^{-1}$ move along the curves $H_{n+1}(L(\rho _n,t))$ , and $H_{n+1}\circ \ell _{\rho _n,t}$ provides a conjugacy between the action of $\varphi _n$ on these curves and the rational rotation $r_{1/q_n}$ . Moreover, if we change the rotation vector $\rho _n$ to $\tilde \rho _n=\alpha \rho _n$ , where $\alpha $ is an irrational real number, then the orbits of $\tilde \varphi _n$ still move along the same curves, but now the action of $\tilde \varphi _n$ on these curves is conjugate to the irrational rotation $r_{\alpha /q_n}$ on $\mathbb {T}^{1}$ (again with conjugacy $H_{n+1}\circ \ell _{\rho _n,t}$ ).

Given two points $z,z'\in \mathbb {T}^2$ with $\pi (z)=\pi (z')$ , we may choose $x,t_z,t_{z'}\in \mathbb {T}^{1}$ such that $z=H_{n+1}(\ell _{\rho _n,t_z}(x))$ and $z'=H_{n+1}(\ell _{\rho _n,t_{z'}}(x))$ . For the average distance of the iterates of these two points along their orbits, we obtain

(4.9) $$ \begin{align} \frac{1}{n}\sum_{i=0}^{n-1}\,d (\tilde\varphi^i_n(z),\tilde\varphi^i_n(z')) & = \frac{1}{n}\sum_{i=0}^{n-1}\,d (\tilde\varphi^i_n (\ell_{\rho_n,t_z}(x)),\tilde\varphi^i_n (\ell_{\rho_n,t_{z'}}(x))) \nonumber\\ & = \frac{1}{n}\sum_{i=0}^{n-1} F_{n,t_z,t_z'}\circ r_{\alpha/q_n}^i(x), \end{align} $$

where

$$ \begin{align*} F_{n,t,t'} : \mathbb{T}^{1}\to\mathbb{R}^+,\quad x\mapsto d (H_{n+1}\circ\ell_{\rho_n,t}(x),\quad H_{n+1}\circ\ell_{\rho_n,t'}(x)). \end{align*} $$

By unique ergodicity of the irrational rotation $r_{\alpha /q_n}$ , the averages in equation (4.9) converge uniformly to $\int _{\mathbb {T}^{1}} F_{n,t,t'}(x)\,dx$ . As the family $\{F_{n,t,t'}\mid t,t'\in \mathbb {T}^{1}\}$ is compact, the simultaneous uniform ergodic theorem 2.3 implies that this convergence is even uniform in the parameters $t,t'\in \mathbb {T}^{1}$ . This means that for any $\kappa>0$ , there exists $K\in \mathbb {N}$ such that

$$ \begin{align*} \bigg| \frac{1}{K}\sum_{i=0}^{K-1}\,d (\tilde\varphi^i_n(z),\tilde\varphi^i_n(z')) - \int_{\mathbb{T}^{1}} F_{n,t_z,t_{z'}}(x)\,ds\bigg| < \kappa \end{align*} $$

holds for all $z,z'\in \mathbb {T}^{1}$ with $\pi (z)=\pi (z')$ . Hence, to ensure the validity of equation (4.6), it suffices to let $\kappa =1/2n$ and to choose $h_{n+1}$ in such a way that

(4.10) $$ \begin{align} \int_{\mathbb{T}^{1}} F_{n,t,t'}(x)\,dx \leq \frac{1}{2n} \end{align} $$

holds for all $t,t'\in \mathbb {T}^{1}$ .

Now, constructing such a mapping $h_{n+1}$ is not difficult, albeit somewhat technical. We first define $h_{n+1}$ on the vertical strip $S=I\times \mathbb {T}^{1}$ , where $I=[0,1/q_n]$ . We fix $\delta>0$ and choose some circle diffeomorphism g, homotopic to the identity, such that $g (\mathbb {T}^{1}\setminus B_\delta (1/2)) \subseteq B_\delta (0)$ . For instance, g could be the projective action of a diagonal matrix $\tiny\begin{pmatrix}\unicode{x3bb} & 0 \\ 0 & 1/\unicode{x3bb} \end {pmatrix}$ with sufficiently large $\unicode{x3bb}>0$ . Then we choose a smooth homotopy $G:[0,1]\times \mathbb {T}^{1}\to \mathbb {T}^{1}$ , $(x,y)\mapsto G_x(y)$ between $G_0=\mathrm {Id}_{\mathbb {T}^{1}}$ and $G_1=g$ such that $G_x=\mathrm {Id}_{\mathbb {T}^{1}}$ for all x in some neighbourhood of zero. We assume $\delta \in [0,1/4q_n]$ , let $\hat I=[\delta ,1/q_n-\delta ]$ and choose a smooth mapping $T:S\to S$ such that $T(x,y)=(x,T_x(y))$ , where $T_x(y)= y + ({x}/({1/q_n-2\delta }))$ for all $(x,y)\in \hat I$ and $T_x=\mathrm {Id}_{\mathbb {T}^{1}}$ for all x in a neighbourhood of zero. Thus, the image of a horizontal line segment $\hat I\times \{y\}$ under T ‘wraps’ around the torus exactly once in the vertical direction.

Using these auxiliary mappings, we let

(4.11) $$ \begin{align} h_{n+1} : S\to S,\quad (x,y)\mapsto \begin{cases} G_{x/\delta}\circ T(x,y), & 0\leq x \leq \delta, \\ g\circ T(x,y), & \delta<x<1/q_n-\delta, \\ G_{(1-x)/\delta}\circ T(x,y), & 1/q_n-\delta \leq x \leq 1/q_n. \end{cases} \end{align} $$

Then $h_{n+1}$ is smooth on S and coincides with the identity on a neighbourhood of $\partial S$ .

Now, we first focus on that segment of a curve $L(\rho _n,t)$ passing through S, which is parametrized by the mapping $p_n^{-1}I\to \mathbb {T}^{1},\ x\mapsto \ell _{\rho _n,t}(x)$ . Note that there are $p_n$ such pieces, which all differ by an additive constant that is a multiple of $1/p_n$ . On $p_n^{-1}\hat I$ , this function has constant slope ${1}/({1/q_n-2c}) + ({p_n}/{q_n})$ . As a consequence, it passes through the set $\hat I\times B_\delta (1/2)$ at most twice over the interval $\hat S$ and we obtain that

(4.12) $$ \begin{align} \mathrm{Leb}_{\mathbb{T}^{1}} (\{ x\in p_n^{-1}I\mid T_x\circ \ell_{\rho_n,t}(x)\in B_\delta(1/2)\}) \leq 2\delta+\frac{\delta}{p_nq_n}. \end{align} $$

Since g maps the complement of the interval $B_\delta (1/2)$ into the interval $B_\delta (0)$ and equation (4.12) holds for all $t\in \mathbb {T}^{1}$ , the images of respective segments of two different curves $L(q_n,t),L(q_n,t')$ under T will be $2\delta $ -close to each other most of the time. More precisely, we obtain

(4.13) $$ \begin{align} \mathrm{Leb}_{\mathbb{T}^{1}}(\{ x\in p_n^{-1}I\mid | h_{n+1,s}\circ \ell_{\rho_n,t}(x)- h_{n+t,s}\circ \ell_{\rho_n,t'}(x)| \geq 2\delta\}) \leq 4\delta+\frac{2\delta}{p_nq_n}. \end{align} $$

So far, we have only defined $h_{n+1}$ on S and only considered the restriction of the curves $\ell _{\rho _n,t}$ to the interval $p_n^{-1}I$ , which only parametrizes the $1/p_nq_n$ th part of the whole curves $L(\rho _n,t)$ . However, if we extend the definition of $h_{n+1}$ by commutativity to all of $\mathbb {T}^2$ , i.e. by setting $h_{n+1|R_{\rho _n}^k(S)}=R_{\rho _n}^k\circ h_{n+1|S}\circ R_{\rho _n}^{-k}$ , $k=1,\ldots , q_n-1$ , then the behaviour of all $p_nq_n$ segments of pairs of curves $h_{n+1}(L(q_n,t))$ and $h_{n+1}(L(q_n,t'))$ will be the same – we are simply looking at a rotated version of the same situation. Therefore, we obtain the estimate

(4.14) $$ \begin{align} \mathrm{Leb}_{\mathbb{T}^{1}} (\{ x\in \mathbb{T}^{1}\mid | h_{n+1,x}\circ \ell_{\rho_n,t}(x)- h_{n+1,x}\circ \ell_{\rho_n,t'}(x)| \geq 2\delta\}) \leq 6\delta p_nq_n. \end{align} $$

Since $H_n$ is uniformly continuous, we may choose $\delta $ in such a way that $d(x,y)<2\delta $ implies $d(H_n(x),H_n(y))< {1}/{4n}$ . Then, equation (4.14) implies

(4.15) $$ \begin{align} \mathrm{Leb}_{\mathbb{T}^{1}} (\{ s\in \mathbb{T}^{1}\mid | H_{n+1,s}\circ \ell_{\rho_n,t}(s)- H_{n+t,s}\circ \ell_{\rho_n,t'}(s)| \geq 1/4n\}) \leq 6\delta p_nq_n. \end{align} $$

When $\delta $ is sufficiently small (say $\delta <1/24np_nq_n$ ), this finally yields equation (4.10).

To complete the induction step, we now choose $\tilde \rho _n=\alpha \rho _n$ , where $\alpha \in \mathbb {R}\setminus \mathbb {Q}$ is sufficiently close to $1$ such that equation (4.3) holds. After that, we can take $\hat \rho _n$ to be any totally irrational rotation vector, close enough to $\tilde \rho _n$ to ensure equation (4.4), and finally choose a new rational rotation vector $\rho _{n+1}$ that is close enough to $\hat \rho _n$ to ensure equation (4.5) and satisfy that equation (4.2) holds. This completes the inductive construction and therefore the proof of Theorem 1.1.

Remark 4.1.

1. (a) The above construction starts with an arbitrary rational rotation $R_{\rho _1}$ and allows to ensure that the resulting limit diffeomorphism $\varphi $ is arbitrarily close to $R_{\rho _1}$ . Together with the $G_\delta $ -property of mean equicontinuity and strict ergodicity, this implies that the set of skew product diffeomorphisms which satisfy the assertions of Theorem 1.1 form a residual subset of the space

   $$ \begin{align*} \overline{\mathrm{Cob}}(\mathbb{T}^2,\pi) = \{ H\circ R_{\rho}\circ H^{-1} \mid \rho\in\mathbb{T}^2,\ H \in \mathrm{Diffeo}^k(\mathbb{T}^2), \pi\circ H=\pi\} , \end{align*} $$
   where $k\in \mathbb {N}_0\cup \{\infty \}$ is arbitrary. The analogous observation has been made in [DG16].

2. (b) All the torus maps in the above construction, and hence also the resulting diffeomorphisms $\varphi $ , may be chosen as the projective actions of quasiperiodic $\textrm {SL}(2,\mathbb {R})$ -cocycles (compare [HP06]).

5. Total strict ergodicity for lifts and non-existence of additional eigenvalues

Given any $l,m\in \mathbb {N}$ , $\mathbb {T}^{(l,m)}=\mathbb {R}/l\mathbb {Z} \times \mathbb {R}/m\mathbb {Z}$ is a canonical finite covering space of the torus $\mathbb {T}^2$ . For any torus homeomorphism $\psi $ homotopic to the identity, there exist lifts $L_{(l,m;s)}^\psi :\mathbb {T}^{(l,m)}\to \mathbb {T}^{(l,m)}$ with $s\in A(l,m)=(\mathbb {Z}/l\mathbb {Z})\times (\mathbb {Z}/m\mathbb {Z})$ , which are uniquely determined by the requirement that $L_{(l,m;s)}^\psi (0)\in [s_1,s_1+1)\times [s_2,s_2+1)$ . Note that two different lifts $L^\psi _{(l,m;s)}$ and $L^\psi _{(l,m;s)}$ are conjugate by the integer translation $(x,y)\mapsto (x+(s_1'-s_1),y+(s_2'-s_s))$ on $\mathbb {T}^{(l,m)}$ and thus share the same dynamical properties.

We denote the iterates of these lifts by $L_{(l,m;s)}^{\psi ,j}=(L_{(l,m;s)}^\psi )^j$ . Note that $L_{(l,m;s)}^{\psi ,j}$ may differ from $L_{(l,m;s)}^{\psi ^j}$ by an integer translation. For any rotation $R_\rho $ on $\mathbb {T}^2$ , its lift $L_{(l,m;s)}^{R_\rho }$ is conjugate to the torus rotation $R_{((\rho _1+s_1)/l,(\rho _2+s_2)/m)}$ , where a conjugacy $h_{(l,m)}$ is simply given by rescaling, that is, $h_{(l,m)}(x,y)=(x/l,y/m)$ . Obviously, this does not affect the arithmetical properties of the rotation vector (rational, irrational or totally irrational). We can also rescale the lifts $L^\psi _{(l,m;s)}$ to obtain homeomorphisms $\ell ^\psi _{(l,m;s)}=h_{(l,m)}\circ L^\psi _{(l,m;s)}\circ h^{-1}_{(l,m)}$ of the standard torus $\mathbb {T}^2$ . Further, given torus homeomorphisms $\varphi ,\psi $ homotopic to the identity, we have

(5.1) $$ \begin{align} d_k(\varphi,\psi) = \min_{s\in A(l,m)}d_k \Big(L^\varphi_{(l,m;0)},L^\psi_{(l,m;s)}\Big) \geq \min_{s\in A(l,m)}d_k \Big(\ell^\varphi_{(l,m;0)},\ell^\psi_{(l,m;s)}\Big). \end{align} $$

5.1. Total strict ergodicity of the lifts

To ensure that all iterates of all lifts of the diffeomorphism $\varphi $ from Theorem 1.1 are strictly ergodic as well, we may now modify the construction in §4 by replacing the conditions in equations (4.3)–(4.5) with the following stronger assumptions: we recursively choose sequences $s^{(l,m)}_n,\tilde s^{(l,m)}_n,\hat s^{(l,m)}_n$ such that

$$ \begin{align*} d_n\Big(L^{\varphi_n}_{( l,m,s^{(l,m)}_n)},L^{\tilde\varphi_n}_{(l,m,\tilde s^{(l,m)}_n)}\Big) & = d_n (\varphi_n,\tilde\varphi_n) , \\ d_n\Big( L^{\tilde\varphi_n}_{( l,m,\tilde s^{(l,m)}_n)}, L^{\hat\varphi_n}_{(l,m,\hat s^{(l,m)}_n)}\Big) & = d_n(\tilde\varphi_n,\hat\varphi_n) , \\ d_n\Big(L^{\hat\varphi_n}_{(l,m,\hat s^{(l,m)}_n)}, L^{\varphi_{n+1}}_{(l,m,s^{(l,m)}_{n+1})}\Big) & = d_n(\hat \varphi_n,\varphi_{n+1}) \end{align*} $$

hold for all $n\in \mathbb {N}$ . Then, in the nth step of the induction, we exert control over the speed of convergence not only for the original maps $\varphi _n,\hat \varphi _n,\tilde \varphi _n$ , but also for all iterates of all lifts up to level n. To that end, we require that for all $l,m,j=1,\ldots , n$ , we have

(5.2) $$ \begin{align} &d_n\Big(\ell^{\varphi_n,j}_{( l,m;s^{(l,m)}_n)},\ell^{\tilde\varphi_{n},j}_{( l,m;\tilde s^{(l,m)}_n)}\Big) \nonumber \\ &\quad \leq \frac{1}{3} \min \Big\{\eta^{\mathrm{se}}_{n-1} \Big(\ell^{\hat\varphi_1,j}_{( l,m;\hat s^{(l,m)}_1)},\ldots, \ell^{\hat\varphi_{n-1},j}_{( l,m;\hat s^{(l,m)}_{n-1})}\Big), \eta^{\mathrm{me}}_{n-1} \Big(\ell^{\tilde\varphi_1,j}_{( l,m;\tilde s^{(l,m)}_1)},\ldots, \ell^{\tilde\varphi_{n-1},j}_{( l,m;\tilde s^{(l,m)}_{n-1})}\Big)\Big\}, \end{align} $$

(5.3) $$ \begin{align} &\hspace{-6pt}{d_n \Big(\ell^{\tilde\varphi_n,j}_{( l,m;\tilde s^{(l,m)}_n)},\ell^{\hat\varphi_{n},j}_{( l,m;\hat s^{(l,m)}_n)}\Big)} \nonumber \\ & \leq \frac{1}{3} \min\Big\{\eta^{\mathrm{se}}_{n-1} \Big(\ell^{\hat\varphi_1,j}_{( l,m;\hat s^{(l,m)}_1)},\ldots, \ell^{\hat\varphi_{n-1},l}_{( l,m;\hat s^{(l,m)}_{n-1})}\Big),\eta^{\mathrm{me}}_{n} \Big(\ell^{\tilde\varphi_1,l}_{( l,m;\tilde s^{(l,m)}_1)},\ldots, \ell^{\tilde\varphi_{n},j}_{( l,m;\tilde s^{(l,m)}_{n})}\Big)\Big\}, \end{align} $$

(5.4) $$ \begin{align} & d_n \Big(\ell^{\hat\varphi_n,j}_{( l,m;\hat s^{(l,m)}_n)},\ell^{\varphi_{n+1},j}_{( l,m;s^{(l,m)}_{n+1})}\Big) \nonumber \\ &\quad \leq \frac{1}{3} \min \Big\{\eta^{\mathrm{se}}_{n} \Big(\ell^{\hat\varphi_1,j}_{( l,m;\hat s^{(l,m)}_1)},\ldots, \ell^{\hat\varphi_{n},j}_{( l,m;\hat s^{(l,m)}_n)}\Big),\eta^{\mathrm{me}}_{n} \Big(\ell^{\tilde\varphi_1,j}_{( l,m;\tilde s^{(l,m)}_1)},\ldots, \ell^{\tilde\varphi_{n},j}_{( l,m;\tilde s^{(l,m)}_n)}\Big)\Big\}. \end{align} $$

With the same reasoning as in §4, we now obtain that for any $(l,m)\in \mathbb {N}^2$ and $j\in \mathbb {N}$ , the sequence $\ell ^{\hat \varphi _n,j}_{(l,m,\hat s^{(l,m)}_n)}$ converges to a strictly ergodic diffeomorphism, which is a rescaled lift $\ell ^{\varphi ,j}_{(l,m,s)}$ of an iterate of $\varphi =\lim _{n\rightarrow \infty } \hat \varphi _n$ .

Let $\psi =\ell _{(1,m;0)}^\varphi $ . The invariant measure of $\varphi $ is of the form $\mu =(\mathrm {Id}_{\mathbb {T}^{1}}\times \gamma )_*\mathrm {Leb}_{\mathbb {T}^{1}}$ , where $\gamma :\mathbb {T}^{1}\to \mathbb {T}^{1}$ is the measurable function whose graph supports $\mu $ . Consequently, if we let

(5.5) $$ \begin{align} \gamma_j:\mathbb{T}^{1}\to\mathbb{T}^{1},\quad x\mapsto \frac{\gamma(x)+j-1}{m},\quad j=1,\ldots, m, \end{align} $$

then

(5.6) $$ \begin{align} \mu^\psi=\frac{1}{m}\sum_{j=1}^m (\mathrm{Id}_{\mathbb{T}^{1}}\times\gamma_j)_*\mathrm{Leb}_{\mathbb{T}^{1}} \end{align} $$

defines an invariant measure for the rescaled lift $\psi $ . By unique ergodicity, it is the only $\psi $ -invariant measure. This proves assertions (a)–(c) of Theorem 1.2.

5.2. Non-existence of additional eigenvalues

We consider a torus diffeomorphism $\varphi $ that satisfies the assertions (a)–(c) of Theorem 1.2. Fix $m\in \mathbb {N}$ and let $\psi =\ell ^\varphi _{(1,m;0)}$ as above. Recall that both mappings are skew products over the irrational rotation $r_\alpha :x\mapsto x+\alpha $ . Our aim is to show that $\psi $ has the same discrete dynamical spectrum as $\varphi $ , that is, there exist no additional dynamical eigenvalues for $\psi $ .

Suppose that $\gamma :\mathbb {T}^{1}\to \mathbb {T}^{1}$ is the measurable function whose graph supports the unique $\varphi $ -invariant measure $\mu $ , that is, $\mu =(\mathrm {Id}_{\mathbb {T}^{1}}\times \gamma )_*\mathrm {Leb}_{\mathbb {T}^{1}}$ . Then, as discussed in the previous section, the unique $\psi $ -invariant measure $\mu ^\psi $ is given by equation (5.6). Now, suppose for a contradiction that $f\in L^2_{\mu ^\psi }(\mathbb {T}^2)$ is an eigenfunction of $U_\psi $ with a new eigenvalue $\unicode{x3bb} $ that is not contained in the group of eigenvalues $M(\alpha )=\{\exp (2\pi ik\alpha )\mid k\in \mathbb {Z}\}$ of $U_\varphi $ . Then f cannot be constant in the fibres (that is, independent of the second coordinate y), since in this case, $x\mapsto f(x,0)$ would define an eigenfunction of $r_\alpha $ with eigenvalue $\unicode{x3bb} $ , contradicting the fact that the eigenvalue group of $r_\alpha $ is $M(\alpha )$ as well. Further, the function

$$ \begin{align*} g:\mathbb{T}^{1}\to\mathbb{T}^{1}, \quad x \mapsto \prod_{j=1}^m f(x,\gamma_j(x)) \end{align*} $$

is an eigenfunction of $r_\alpha $ with eigenvalue $\unicode{x3bb} ^m$ , since we have

$$ \begin{align*} \psi(\{(x,\gamma_1(x)),\ldots, (x,\gamma_m(x))\}) = \{(x+\alpha,\gamma_1(x+\alpha)),\ldots, (x+\alpha,\gamma_m(x+\alpha))\} \end{align*} $$

and therefore

$$ \begin{align*} g(x+\alpha) = \prod_{j=1}^m f\circ\psi(x,\gamma_j(x)) = \unicode{x3bb}^m\prod_{j=1}^m f(x,\gamma_j(x)) = \unicode{x3bb}^m g(x) \end{align*} $$

$\mathrm {Leb}_{\mathbb {T}^{1}}$ -almost surely on $\mathbb {T}^{1}$ . Hence, g is an eigenfunction of $r_\alpha $ . This implies $\unicode{x3bb} ^m=\exp (2\pi ik\alpha )$ for some $k\in \mathbb {Z}$ , so that $\unicode{x3bb} $ must be of the form

$$ \begin{align*} \unicode{x3bb} = \exp \bigg(2\pi i\bigg(\frac{k\alpha+p}{m}\bigg)\bigg) \end{align*} $$

for some $(k,p)\in (\mathbb {Z}\times \{0,\ldots , m-1\})\setminus (m\mathbb {Z}\times \{0\})$ .

We first assume that $k=0$ . In this case, f is an eigenfunction of $\psi ^m$ with eigenvalue $\unicode{x3bb} ^m=1$ . As f is non-constant, this contradicts the ergodicity of $\psi ^m$ .

Second, assume that $p=0$ . In this case, we consider the rescaled lift $\tilde \psi =\ell ^\psi _{(1,m,0)}$ of $\psi $ , which is now a skew product over the rotation $r_{\alpha /m}$ . The eigenfunction f transforms to an eigenfunction

$$ \begin{align*} \tilde f(x,y) = f(mx,y) \end{align*} $$

of $U_{\tilde \psi }$ , which still has the same eigenvalue $\unicode{x3bb} =\exp (2\pi ik\alpha /m)$ . However, this is now an eigenvalue of the underlying rotation $r_{\alpha /m}$ , which corresponds to the eigenfunction $g(x,y)=\exp (2\pi ikx/m)$ of $U_{\tilde \psi }$ . As g is constant in y for $\nu $ -almost every x, but $\tilde f$ is not, the two eigenfunctions cannot coincide. Since they have the same eigenvalue, this contradicts the unique ergodicity of $\tilde \psi $ (note that $\tilde \psi $ is still a lift of the original map $\varphi $ and is therefore uniquely ergodic).

Finally, we consider the case where $k\neq 0\neq p$ . In this case, f is an eigenfunction of $\psi ^m$ , with eigenvalue $\exp (2\pi ik\alpha )$ . However, $\psi ^m$ is a rescaled lift of $\varphi ^m$ , which has the same properties as $\varphi $ (it satisfies the assertions of Theorem 1.1), but has underlying rotation number $m\alpha $ . This means that we are in exactly the same situation as in the case $p=0$ above, and again arrive at a contradiction.

Altogether, this shows that $\psi $ has exactly the same dynamical eigenvalues as $\varphi $ . However, the two systems cannot be isomorphic, as measure-theoretic factor maps into group rotations are uniquely determined up to post-composition with a rotation and the canonical factor map from $\psi $ to $\varphi $ is $m:1$ . Due to the Halmos–von Neumann theorem, $\psi $ and $\varphi $ cannot have the same (purely discrete) dynamical spectrum. This means that the spectrum of $U_\psi $ must have a continuous component.

5.3. Singularity of the continuous spectral component

To complete the proof of Theorem 1.2, our aim now is to show that the Anosov–Katok construction of $\varphi $ can be modified such that the map $\psi $ defined in the last section has a singular continuous spectral component. To that end, we need to show that $\varphi $ and all its lifts admit cyclic approximation by periodic transformations with speed $o(1/n)$ , in the sense of Theorem 2.5. The main problem here lies in the fact that – unlike for Anosov–Katok construction in an area-preserving setting – the unique invariant measure $\mu $ of the transformation $\varphi $ is not known a priori. Therefore, it is necessary to control both the size of the symmetric differences between the images of partition elements under $\varphi _n$ and the eventual limit $\varphi $ and also the limit measure of these sets at the same time. Recall that, in the end, we need to show that there exist suitable partitions $\mathcal {P}_n$ that satisfy conditions (P1)–(P3) from §2.2. This will exclude the existence of an absolutely continuous spectral component and thus complete the proof.

We adopt the notation from the main construction in §4. In particular, $q_n$ is the denominator of the rotation vector $\rho _n$ of the nth approximating diffeomorphism $\varphi _n$ . Note that $\rho _n$ was chosen only after the nth conjugating diffeomorphism $H_n$ was defined. Hence, we can require that

(5.7) $$ \begin{align} d(x,y)<2/q_n \Rightarrow d(H_n(x),H_n(y))<1/n. \end{align} $$

We define the partition $\mathcal {P}_n$ as $\mathcal {P}_n=\{P_{n,i,j}\mid i,j=0,\ldots , q_n-1\}$ , where

$$ \begin{align*} P_{n,i,j}=H_n ([i/q_n,(i+1)/q_n)\times [j/q_n,(j+1)/q_n)). \end{align*} $$

Note that $\varphi _n=H_n\circ R_{\rho _n}\circ H_n^{-1}$ cyclically permutes the elements of $\mathcal {P}_n$ due to the fact that $\rho _n=(p_n/q_n,p_n'/q_n)$ with $p_n,p_n',q_n$ relatively prime in equation (4.2). Moreover, due to equation (5.7), the maximal diameter of an element of $\mathcal {P}_n$ is at most $1/n$ , which implies condition (P2) due to the regularity of the measure $\mu $ . Hence, both conditions (P1) and (P2) are satisfied.

It remains to show condition (P3) with sufficiently fast speed of convergence. We choose an arbitrary function $s:\mathbb {N}\to \mathbb {R}^+$ which satisfies $\lim _{n\rightarrow \infty } ns(n)=0$ , so that Theorem 2.5 will be applicable. As $K_n=\sharp \mathcal {P}_n=q_n^2$ , we have to ensure that

(5.8) $$ \begin{align} \sum_{i,j=0}^{q_n-1} \mu ( \varphi_n ( P_{n,i,j}) \triangle \varphi (P_{n,i,j})) \leq s(q_n^2). \end{align} $$

To do so, we need to introduce further inductive assumptions into the construction carried out in §4 that we already modified by equations (5.2)–(5.4) above.

Suppose that $n\in \mathbb {N}$ and $H_{n+1}$ has already been chosen, but not the rotation vectors $\tilde \rho _n,\ \hat \rho _n$ and $\rho _{n+1}$ (which then define $\tilde \varphi _n,\ \hat \varphi _n$ and $\varphi _{n+1}$ ). Let $\mu _n= (H_{n+1})_*\mathrm {Leb}_{\mathbb {T}^2}$ and note that, independent of the choice $\hat \rho _n$ (assuming total irrationality), this is the unique invariant measure of $\hat \varphi _n=H_{n+1}\circ R_{\rho _n}\circ H^{-1}_{n+1}$ . For $i,j=1,\ldots , q_n$ , we choose continuous functions $f_{n,i,j}:\mathbb {T}^2\to [0,1]$ such that

$$ \begin{align*} \partial (\varphi_n(P_{n,i,j}))\subseteq \mathrm{int}(f_{n,i,j}^{-1}(1)) \end{align*} $$

and

$$ \begin{align*} \int_{\mathbb{T}^2} f_{n,i,j}\,d\mu_{n} < s(q_n^2)/q_n^2. \end{align*} $$

For the latter condition, note that since $\varphi _n$ simply permutes the elements of $\mathcal {P}_n$ , the set $\partial (\varphi _n(P_{n,i,j}))$ is simply the boundary of another partition element, and therefore a smooth curve that has measure zero with respect to $\mu _n$ (which has smooth density with respect to Lebesgue, since it is the image of the Lebesgue measure under the smooth diffeomorphism $H_n$ ).

If $\tilde \rho _n$ and subsequently $\hat \rho _n$ are chosen sufficiently close to $\rho _n$ , so that $\tilde \varphi _n$ and $\hat \varphi _n$ are close to $\varphi _n$ , then we have

(5.9) $$ \begin{align} \hat\varphi_n (P_{n,i,j}) \Delta \varphi_n (P_{n,i,j}) \subseteq \mathrm{int}(f_{n,i,j}^{-1}(1)). \end{align} $$

By unique ergodicity, there exists $M_n\in \mathbb {N}$ such that

(5.10) $$ \begin{align} \sup_{x\in\mathbb{T}^2} \frac{1}{M_n} \sum_{l=1}^{M_n} f_{n,i,j}\circ \hat\varphi_n^l(x) < s(q_n^2)/q_n^2. \end{align} $$

Since both equations (5.9) and (5.10) are open conditions, we may now choose $\delta _n>0$ such that, for all $\psi \in \overline {B_{\delta _n}(\hat \varphi _n)}$ , the following conditions hold:

(5.11) $$ \begin{align} \psi (P_{n,i,j}) \Delta \varphi_n (P_{n,i,j}) & \subseteq \mathrm{int}(f_{n,i,j}^{-1}(1)) \end{align} $$

(5.12) $$ \begin{align} \sup_{x\in\mathbb{T}^2} \frac{1}{M_n} \sum_{l=1}^{M_n} f_{n,i,j}\circ \psi^l(x) & < s(q_n^2)/q_n^2. \end{align} $$

We can now require, throughout the inductive construction in §4 that for all $n\in \mathbb {N}$ , we have

(5.13) $$ \begin{align} d_0 (\hat\varphi_n,\hat\varphi_m) \leq \delta_m \quad\textrm{for all } m=1,\ldots, n-1. \end{align} $$

For this, when going from n to $n+1$ , it suffices to ensure that

$$ \begin{align*} \max \{ d_0 (\hat\varphi_n,\varphi_{n+1}), d_0 (\varphi_{n+1},\tilde\varphi_{n+1}), d_0(\tilde\varphi_{n+1},\hat\varphi_{n+1})\} < \frac{1}{3} \min_{m=1}^n \delta_m-d_0 (\hat\varphi_n,\hat\varphi_m). \end{align*} $$

This, in turn, is simply achieved by a sufficiently small variation of the rotation vectors when choosing $\rho _{n+1},\tilde \rho _{n+1}$ and $\hat \rho _{n+1}$ . In particular, it does not contradict any other recursive assumptions that we have made elsewhere during the construction.

As a consequence, the resulting limit $\varphi $ will still satisfy equations (5.11) and (5.12) (with $\psi $ replaced by $\varphi $ ). However, if $\mu $ denotes the unique $\varphi $ -invariant measure, then the above conditions imply that, for all $n\in \mathbb {N}$ ,

$$ \begin{align*} \mu (\varphi (P_{n,i,j}) \Delta \varphi_n (P_{n,i,j}) ) \stackrel{(5.11)}{\leq} \int_{\mathbb{T}^2} f_{n,i,j}\,d\mu \stackrel{(5.12)}{\leq} s(q_n^2)/q_n^2. \end{align*} $$

This proves equation (5.8), so that Theorem 2.5 yields the absence of singular continuous spectrum for $U_\varphi $ . Hence, assertion (d) of Theorem 1.2 holds, which completes the proof.