We establish a ‘diagonal’ ergodic theorem involving the additive and multiplicative groups of a countable field $K$ and, with the help of a new variant of Furstenberg’s correspondence principle, prove that any ‘large’ set in $K$ contains many configurations of the form $\{x+y,xy\}$. We also show that for any finite coloring of $K$ there are many $x,y\in K$ such that $x,x+y$ and $xy$ have the same color. Finally, by utilizing a finitistic version of our main ergodic theorem, we obtain combinatorial results pertaining to finite fields. In particular, we obtain an alternative proof for a result obtained by Cilleruelo [Combinatorial problems in finite fields and Sidon sets. Combinatorica32(5) (2012), 497–511], showing that for any finite field $F$ and any subsets $E_{1},E_{2}\subset F$ with $|E_{1}|\,|E_{2}|>6|F|$, there exist $u,v\in F$ such that $u+v\in E_{1}$ and $uv\in E_{2}$.

We investigate the shifts associated with natural codings of linear involutions. We deduce, from the geometric representation of linear involutions as Poincaré maps of measured foliations, a suitable definition of return words which yields that the set of return words to a given word is a symmetric basis of the free group on the underlying alphabet, $A$ . The set of return words with respect to a subgroup of finite index $G$ of the free group on $A$ is also proved to be a symmetric basis of $G$ .

Given a relatively prime pair of integers, $n\geq m>1$, there is associated a topological dynamical system which we refer to as an $n/m$-solenoid. It is also a Smale space, as defined by David Ruelle, meaning that it has local coordinates of contracting and expanding directions. In this case, these are locally products of the real and various $p$-adic numbers. In the special case, $m=2,n=3$ and for $n>3m$, we construct Markov partitions for such systems. The second author has developed a homology theory for Smale spaces and we compute this in these examples, using the given Markov partitions, for all values of $n\geq m>1$ and relatively prime.

We give explicit formulae for fringe lengths of the Calegari–Walker ziggurats—i.e., graphs of extremal rotation numbers associated with positive words in free groups. These formulae reveal (partial) integral projective self-similarity in ziggurat fringes, which are low-dimensional projections of characteristic polyhedra on the bounded cohomology of free groups. This explains phenomena observed experimentally by Gordenko, Calegari and Walker.

We consider a non-uniquely ergodic dynamical system given by a $\mathbb{Z}^{l}$-action (or $(\mathbb{N}\cup \{0\})^{l}$-action) $\unicode[STIX]{x1D70F}$ on a non-empty compact metrisable space $\unicode[STIX]{x1D6FA}$, for some $l\in \mathbb{N}$. Let (D) denote the following property: the graph of the restriction of the entropy map $h^{\unicode[STIX]{x1D70F}}$ to the set of ergodic states is dense in the graph of $h^{\unicode[STIX]{x1D70F}}$. We assume that $h^{\unicode[STIX]{x1D70F}}$ is finite and upper semi-continuous. We give several criteria in order that (D) holds, each of which is stated in terms of a basic notion: Gateaux differentiability of the pressure map $P^{\unicode[STIX]{x1D70F}}$ on some sets dense in the space $C(\unicode[STIX]{x1D6FA})$ of real-valued continuous functions on $\unicode[STIX]{x1D6FA}$, level-two large deviation principle, level-one large deviation principle, convexity properties of some maps on $\mathbb{R}^{n}$ for all $n\in \mathbb{N}$. The one involving the Gateaux differentiability of $P^{\unicode[STIX]{x1D70F}}$ is of particular relevance in the context of large deviations since it establishes a clear comparison with another well-known sufficient condition: we show that for each non-empty $\unicode[STIX]{x1D70E}$-compact subset $\unicode[STIX]{x1D6F4}$ of $C(\unicode[STIX]{x1D6FA})$, (D) is equivalent to the existence of an infinite dimensional vector space $V$ dense in $C(\unicode[STIX]{x1D6FA})$ such that $f+g$ has a unique equilibrium state for all $(f,g)\in \unicode[STIX]{x1D6F4}\times V\setminus \{0\}$; any Schauder basis $(f_{n})$ of $C(\unicode[STIX]{x1D6FA})$ whose linear span contains $\unicode[STIX]{x1D6F4}$ admits an arbitrary small perturbation $(h_{n})$ so that one can take $V=\text{span}(\{f_{n}+h_{n}:n\in \mathbb{N}\})$. Taking $\unicode[STIX]{x1D6F4}=\{0\}$, the existence of an infinite dimensional vector space dense in $C(\unicode[STIX]{x1D6FA})$ constituted by functions admitting a unique equilibrium state is equivalent to (D) together with the uniqueness of the measure of maximum entropy.

The Fischer, Krieger, and fiber product covers of sofic beta-shifts are constructed and used to show that every strictly sofic beta-shift is 2-sofic. Flow invariants based on the covers are computed, and shown to depend only on a single integer that can easily be determined from the $\unicode[STIX]{x1D6FD}$-expansion of 1. It is shown that any beta-shift is flow equivalent to a beta-shift given by some $1<\unicode[STIX]{x1D6FD}<2$, and concrete constructions lead to further reductions of the flow classification problem. For each sofic beta-shift, there is an action of $\mathbb{Z}/2\mathbb{Z}$ on the edge shift given by the fiber product, and it is shown precisely when there exists a flow equivalence respecting these $\mathbb{Z}/2\mathbb{Z}$-actions. This opens a connection to ongoing efforts to classify general irreducible 2-sofic shifts via flow equivalences of reducible shifts of finite type (SFTs) equipped with $\mathbb{Z}/2\mathbb{Z}$-actions.

We propose a new criterion for randomness of a word $x_{1}x_{2}\cdots x_{n}\in \mathbb{A}^{n}$ over a finite alphabet $\mathbb{A}$ defined by

$$\begin{eqnarray}\unicode[STIX]{x1D6EF}^{n}(x_{1}x_{2}\cdots x_{n})=\mathop{\sum }_{\unicode[STIX]{x1D709}\in \mathbb{A}^{+}}\unicode[STIX]{x1D713}(|x_{1}x_{2}\cdots x_{n}|_{\unicode[STIX]{x1D709}}),\end{eqnarray}$$

$\mathbb{A}^{+}=\bigcup _{k=1}^{\infty }\mathbb{A}^{k}$

$\mathbb{A}$

$\unicode[STIX]{x1D709}\in \mathbb{A}^{k}$

$$\begin{eqnarray}|x_{1}x_{2}\cdots x_{n}|_{\unicode[STIX]{x1D709}}=\#\{i;~1\leq i\leq n-k+1,~x_{i}x_{i+1}\cdots x_{i+k-1}=\unicode[STIX]{x1D709}\},\end{eqnarray}$$

$t\geq 0$

$\unicode[STIX]{x1D713}(0)=0$

$\unicode[STIX]{x1D713}(t)=t\log t~(t>0)$

$x_{1}x_{2}\cdots x_{n}$

$$\begin{eqnarray}Q(x_{1}x_{2}\cdots x_{n})=(1/2)(n\log n)^{2}/\unicode[STIX]{x1D6EF}^{n}(x_{1}x_{2}\cdots x_{n}).\end{eqnarray}$$

$$\begin{eqnarray}\lim _{n\rightarrow \infty }(1/n)Q(X_{1}X_{2}\cdots X_{n})=h(X)\end{eqnarray}$$

$X_{1}X_{2}\cdots \,$

$\mathbb{A}$

$h(X)=0$

$h(X)$

$t^{2}$

$t\log t$

The critical dimension of an ergodic non-singular dynamical system is the asymptotic growth rate of sums of consecutive Radon–Nikodým derivatives. This has been shown to equal the average coordinate entropy for product odometers when the size of individual factors is bounded. We extend this result to $G$-measures with an asymptotic bound on the size of individual factors. Furthermore, unlike von Neumann–Krieger type, the critical dimension is an invariant property on the class of ergodic $G$-measures.

This paper considers the general question of when a topological action of a countable group can be factored into a direct product of non-trivial actions. In the early 1980s, D. Lind considered such questions for $\mathbb{Z}$-shifts of finite type. In particular, we study direct factorizations of subshifts of finite type over $\mathbb{Z}^{d}$ and other groups, and $\mathbb{Z}$-subshifts which are not of finite type. The main results concern direct factors of the multidimensional full $n$-shift, the multidimensional $3$-colored chessboard and the Dyck shift over a prime alphabet. A direct factorization of an expansive $\mathbb{G}$-action must be finite, but an example is provided of a non-expansive $\mathbb{Z}$-action for which there is no finite direct-prime factorization. The question about existence of direct-prime factorization of expansive actions remains open, even for $\mathbb{G}=\mathbb{Z}$.

The multicorns are the connectedness loci of unicritical antiholomorphic polynomials $\bar{z}^{d}+c$. We investigate the structure of boundaries of hyperbolic components: we prove that the structure of bifurcations from hyperbolic components of even period is as one would expect for maps that depend holomorphically on a complex parameter (for instance, as for the Mandelbrot set; in this setting, this is a non-obvious fact), while the bifurcation structure at hyperbolic components of odd period is very different. In particular, the boundaries of odd period hyperbolic components consist only of parabolic parameters, and there are bifurcations between hyperbolic components along entire arcs, but only of bifurcation ratio 2. We also count the number of hyperbolic components of any period of the multicorns. Since antiholomorphic polynomials depend only real-analytically on the parameters, most of the techniques used in this paper are quite different from the ones used to prove the corresponding results in a holomorphic setting.

Let $D^{-}$ and $D^{+}$ be properly immersed closed locally convex subsets of a Riemannian manifold with pinched negative sectional curvature. Using mixing properties of the geodesic flow, we give an asymptotic formula as $t\rightarrow +\infty$ for the number of common perpendiculars of length at most $t$ from $D^{-}$ to $D^{+}$, counted with multiplicities, and we prove the equidistribution in the outer and inner unit normal bundles of $D^{-}$ and $D^{+}$ of the tangent vectors at the endpoints of the common perpendiculars. When the manifold is compact with exponential decay of correlations or arithmetic with finite volume, we give an error term for the asymptotic. As an application, we give an asymptotic formula for the number of connected components of the domain of discontinuity of Kleinian groups as their diameter goes to $0$.

Let $X$ be a proper, geodesically complete CAT($0$) space under a proper, non-elementary, isometric action by a group $\unicode[STIX]{x1D6E4}$ with a rank one element. We construct a generalized Bowen–Margulis measure on the space of unit-speed parametrized geodesics of $X$ modulo the $\unicode[STIX]{x1D6E4}$-action. Although the construction of Bowen–Margulis measures for rank one non-positively curved manifolds and for CAT($-1$) spaces is well known, the construction for CAT($0$) spaces hinges on establishing a new structural result of independent interest: almost no geodesic, under the Bowen–Margulis measure, bounds a flat strip of any positive width. We also show that almost every point in $\unicode[STIX]{x2202}_{\infty }X$, under the Patterson–Sullivan measure, is isolated in the Tits metric. (For these results we assume the Bowen–Margulis measure is finite, as it is in the cocompact case.) Finally, we precisely characterize mixing when $X$ has full limit set: a finite Bowen–Margulis measure is not mixing under the geodesic flow precisely when $X$ is a tree with all edge lengths in $c\mathbb{Z}$ for some $c>0$. This characterization is new, even in the setting of CAT($-1$) spaces. More general (technical) versions of these results are also stated in the paper.

We apply coupling techniques in order to prove that the transfer operators associated with random topological Markov chains and non-stationary shift spaces with the big images and preimages property have a spectral gap.

Irreducible interval exchange transformations are studied with regard to the whirly property, a condition for a non-trivial spatial factor. A uniformly whirly transformation is defined and is further studied. An equivalent condition is introduced for the whirly transformation. We will prove that almost all 3-interval exchange transformations are whirly, using a combinatorics approach with application of the Rauzy–Veech induction. It is still an open question whether the whirly property is a generic property for $m$-interval exchange transformations for $m\geq 4$.