We show that among three-interval exchange transformations there exists a dichotomy: T has minimal self-joinings whenever the associated subshift is linearly recurrent, and is rigid otherwise. We also build a family of simple rigid three-interval exchange transformations, which is a step towards an old question of Veech, and a family of rigid three-interval exchange transformations, which includes Katok's rank-one map.

Let denote the orbit closure, under the left shift σ, of the sequence… (all zeroes)… 101000101000000000101 … corresponding to the integer Cantor set . We prove that with respect to the infinite invariant measure ρ, which is the unique normalized non-atomic invariant measure on M, for every f ∈ L1(M, ρ), for ρ-a.e. x∈ M

where d = log 2/log 3, and c is the almost-sure value of the right-hand order-two density of the middle-third Cantor set. The proof uses renormalization to a scaling flow, plus identification of (M, σ) as a tower over the Kakutani-von Neumann dyadic odometer.

For locally compact groups all actions on a standard measure algebra have a spatial realization. For many Polish groups this is no longer the case. However, we show here that for non-archimedean Polish groups all measure algebra actions do have spatial realizations. In the other direction we show that an action of a Polish group is whirly (‘ergodic at the identity’) if and only if it admits no spatial factors and that all actions of a Lévy group are whirly. We also show that in the Polish group $\rm{Aut\,}(X,\mathcal{X},\mu)$, for the generic automorphism T the action of the subgroup $\Lambda(T)=\rm{cls\,}\{T^n: n\in \mathbb{Z}\}$ on the Lebesgue space $(X,\mathcal{X},\mu)$ is whirly.

Let $\Gamma$ denote a word-hyperbolic group, and let $S=S^{-1}$ denote a finite symmetric set of generators. Let $S_n = \{w: |w| = n \}$ denote the sphere of radius $n$, where $|\cdot|$ denotes the word length on $\Gamma$ induced by $S$. Define $\s_n \deq (1/{\#S_n}) \sum_{w \in S_n}w$, and $\mu_n=(1/(n+1))\sum_{k=0}^n \s_k$. Let $(X,{\cal B},m)$ be a probability space on which $\Gamma$ acts ergodically by measure-preserving transformations. We prove a strong maximal inequality in $L^2$ for the maximal operator $f_{\mu}^*=\sup_{n\ge 0}|\mu_n f(x)|$. The maximal inequality is applied to prove a pointwise ergodic theorem in $L^2$ for exponentially mixing actions of $\Gamma$ of the following form: $\mu_n f(x) \to \int_X f\,dm$ almost everywhere and in the $L^2$-norm, for every $f\in L^2(X)$. As a corollary, for a uniform lattice $\Gamma \subset G$, where $G$ is a simple Lie group of real rank one, we obtain a pointwise ergodic theorem for the action of $\Gamma$ on an arbitrary ergodic $G$-space. In particular, this result holds when $X=G/\Lambda$ is a compact homogeneous space, and yields an equidistribution result for sets of lattice points of the form $\Gamma g$, for almost every $g\in G$.

It is known that the positiveness of topological entropy, the existence of a horseshoe and the existence of a homoclinic trajectory are mutually equivalent, for interval maps. The aim of the paper is to investigate the relations between the properties for continuous maps of trees, graphs and dendrites. We consider three different definitions of a horseshoe and two different definitions of a homoclinic trajectory. All the properties are mutually equivalent for tree maps, whereas not for maps on graphs and dendrites. For example, positive topological entropy and the existence of a homoclinic trajectory are independent and neither of them implies the existence of any horseshoe in the case of dendrites. Unfortunately, there is still an open problem, and we formulate it at the end of the paper.

Cyclic algebraic ${\mathbb {Z}^{d}}$-actions are defined by ideals of Laurent polynomials in $d$ commuting variables. Such an action is expansive precisely when the complex variety of the ideal is disjoint from the multiplicative $d$-torus. For such expansive actions it is known that the limit of the growth rate of periodic points exists and is equal to the entropy of the action. In an earlier paper the authors extended this result to ideals whose variety intersects the $d$-torus in a finite set. Here we further extend it to the case where the dimension of intersection of the variety with the $d$-torus is at most $d-2$. The main tool is the construction of homoclinic points which decay rapidly enough to be summable.

This paper deals with families of conformal iterated function systems (CIFS). The space of all CIFS, with common seed space X and alphabet I, is successively endowed with the topology of pointwise convergence and a new, weaker topology called $\lambda$-topology. It is proved that the pressure and the Hausdorff dimension of the limit set are continuous with respect to the topology of pointwise convergence when I is finite, and are lower semi-continuous, though generally not continuous, when I is infinite. It is then shown that these two functions are, in any case, continuous in the $\lambda$-topology. The concepts of analytic, regularly analytic and plane-analytic families of CIFS are also introduced. It is established that if a family of CIFS is regularly analytic, then the Hausdorff dimension function is real-analytic; if a family is plane-analytic, then the Hausdorff dimension function is continuous and subharmonic, though not necessarily real-analytic. These results are then applied to finite parabolic CIFS. Counter-examples highlighting breakdowns of real-analyticity in the Hausdorff dimension among analytic, but not regularly analytic, families are further provided. Such families often exhibit a phenomenon known as phase transition. Sufficient conditions preventing the occurrence of such transitions are supplemented.

This paper generalises the well known structural stability theorem which says that a diffeomorphism is structurally stable if it is Axiom A and if all invariant manifolds are transversal to each other.

If these transversality conditions are not satisfied then the diffeomorphism not only fails to be stable, but also this gives rise to the appearance of moduli. That is, one needs several real parameters to parameterise all conjugacy classes of nearby diffeomorphisms. (The minimum number of parameters needed is called the number of moduli).

Here we deal with diffeomorphisms on two dimensional manifolds, whose asymptotic dynamics are well understood (the class of Axiom A diffeomorphisms). The main result characterises those Axiom A diffeomorphisms which have a finite number of moduli. This result can be regarded as a generalisation of the structural stability theorem. From the proofs it follows that the dynamics of these diffeomorphisms can also be well understood.

In the proof of our main theorem we need certain invariant foliations to be quite smooth. In an appendix we prove a differentiable version of the Lambda Lemma.

We consider Hölder continuous fiber bunched $\text{GL}(d,\mathbb{R})$-valued cocycles over an Anosov diffeomorphism. We show that two such cocycles are Hölder continuously cohomologous if they have equal periodic data, and prove a result for cocycles with conjugate periodic data. We obtain a corollary for cohomology between any constant cocycle and its small perturbation. The fiber bunching condition means that non-conformality of the cocycle is dominated by the expansion and contraction in the base. We show that this condition can be established based on the periodic data. Some important examples of cocycles come from the differential of a diffeomorphism and its restrictions to invariant sub-bundles. We discuss an application of our results to the question of whether an Anosov diffeomorphism is smoothly conjugate to a $C^{1}$-small perturbation. We also establish Hölder continuity of a measurable conjugacy between a fiber bunched cocycle and a uniformly quasiconformal one. Our main results also hold for cocycles with values in a closed subgroup of $\text{GL}(d,\mathbb{R})$, for cocycles over hyperbolic sets and shifts of finite type, and for linear cocycles on a non-trivial vector bundle.

We consider the dynamics on the $C^{\ast }$-algebras of finite graphs obtained by lifting the gauge action to an action of the real line. Enomoto, Fujii and Watatani [KMS states for gauge action on ${\mathcal{O}}_{A}$. Math. Japon.29 (1984), 607–619] proved that if the vertex matrix of the graph is irreducible, then the dynamics on the graph algebra admits a single Kubo–Martin–Schwinger (KMS) state. We have previously studied the dynamics on the Toeplitz algebra, and explicitly described a finite-dimensional simplex of KMS states for inverse temperatures above a critical value. Here we study the KMS states for graphs with reducible vertex matrix, and for inverse temperatures at and below the critical value. We prove a general result which describes all the KMS states at a fixed inverse temperature, and then apply this theorem to a variety of examples. We find that there can be many patterns of phase transition, depending on the behaviour of paths in the underlying graph.

We show that a partially hyperbolic $C^{1}$-diffeomorphism $f:M\rightarrow M$ with a uniformly compact $f$-invariant center foliation ${\mathcal{F}}^{c}$ is dynamically coherent. Further, the induced homeomorphism $F:M/{\mathcal{F}}^{c}\rightarrow M/{\mathcal{F}}^{c}$ on the quotient space of the center foliation has the shadowing property, i.e. for every ${\it\epsilon}>0$ there exists ${\it\delta}>0$ such that every ${\it\delta}$-pseudo-orbit of center leaves is ${\it\epsilon}$-shadowed by an orbit of center leaves. Although the shadowing orbit is not necessarily unique, we prove the density of periodic center leaves inside the chain recurrent set of the quotient dynamics. Other interesting properties of the quotient dynamics are also discussed.

We consider the action of the (n-1)-dimensional group of diagonal matrices in SL(n,\mathbb{R}) on SL(n,\mathbb{R})/\Gamma, where \Gamma is a lattice and n\ge 3. Far-reaching conjectures of Furstenberg, Katok–Spatzier and Margulis suggest that there are very few closed invariant sets for this action. We examine the closed invariant sets containing compact orbits. For example, for \Gamma={\rm SL}(n,\mathbb{Z}) we describe all possible orbit-closures containing a compact orbit. In marked contrast to the case n=2, such orbit-closures are necessarily homogeneous submanifolds in the sense of Ratner.

Let $f:M\to M$ be a partially hyperbolic diffeomorphism, ${\it TM}=E^{ss}\oplus E^c\oplus E^{uu}$ such that the stable foliation $\mathcal{F}^{ss}(f)$ is minimal. We give a sufficient condition so that this foliation remains minimal after perturbations, i.e. $\mathcal{F}^{ss}(g)$ is minimal for every g sufficiently close to f.

This paper surveys the work of Charles Conley and his students on Morse decompositions for flows on compact metric spaces, as well as the more recent development of the connection matrix formalism for detecting connections between the Morse sets of a Morse decomposition.

Given a ${\cal C}^{1+\gamma}$ hyperbolic Cantor set $C$, we study the sequence $C_{n,x}$ of Cantor subsets which nest down toward a point $x$ in $C$. We show that $C_{n,x}$ is asymptotically equal to an ergodic Cantor set valued process. The values of this process, called limit sets, are indexed by a Hölder continuous set-valued function defined on Sullivan's dual Cantor set. We show the limit sets are themselves ${\cal C}^{k+\gamma},{\cal C}^\infty$ or ${\cal C}^\omega$ hyperbolic Cantor sets, with the highest degree of smoothness which occurs in the ${\cal C}^{1+\gamma}$ conjugacy class of $C$. The proof of this leads to the following rigidity theorem: if two ${\cal C}^{k+\gamma},{\cal C}^\infty$ or ${\cal C}^\omega$ hyperbolic Cantor sets are ${\cal C}^1$ conjugate, then the conjugacy (with a different extension) is in fact already ${\cal C}^{k+\gamma},{\cal C}^\infty$ or ${\cal C}^\omega$. Within one ${\cal C}^{1+\gamma}$ conjugacy class, each smoothness class is a Banach manifold, which is acted on by the semigroup given by rescaling subintervals. Smoothness classes nest down, and contained in the intersection of them all is a compact set which is the attractor for the semigroup: the collection of limit sets. Convergence is exponentially fast, in the ${\cal C}^1$ norm.

In this article we prove that given a self-similar interval exchange transformation T(λ,π), whose associated matrix verifies a quite general algebraic condition, there exists an affine interval exchange transformation with wandering intervals that is semi-conjugated to it. That is, in this context the existence of Denjoy counterexamples occurs very often, generalizing the result of Cobo [Piece-wise affine maps conjugate to interval exchanges. Ergod. Th. & Dynam. Sys.22 (2002), 375–407].

We present here an investigation of the degree to which mixing properties can be shown to lift to cocycle extensions of an ergodic map by ℤn and ℝn weakly mixing actions. A number of other results on such extensions are also included.

We prove that any finitely generated solvable group of zero entropy contains a nilpotent subgroup of finite index. In particular, any finitely generated solvable group of exponential growth is of uniformly exponential growth.

In this paper we show that the recent work of Dolgopiat and earlier work of the author give a rich class of examples of Axiom A attracting flows which mix exponentially fast. As an immediate corollary we have a proof of a conjecture of Ruelle.

We have two main purposes in this paper. One is to give some sufficient conditions for the Julia set of a transcendental entire function $f$ to be connected or to be disconnected as a subset of the complex plane ${\Bbb C}$. The other is to investigate the boundary of an unbounded periodic Fatou component $U$, which is known to be simply-connected. These are related as follows: let $\varphi : {\Bbb D} \longrightarrow U$ be a Riemann map of $U$ from a unit disk ${\Bbb D}$, then under some mild conditions we show that the set $\Theta_{\infty}$ of all angles where $\varphi$ admits the radial limit $\infty$ is dense in $\partial {\Bbb D}$ if $U$ is an attracting basin, a parabolic basin or a Siegel disk. If $U$ is a Baker domain on which $f$ is not univalent, then $\Theta_{\infty}$ is dense in $\partial {\Bbb D}$ or at least its closure $\overline{\Theta_{\infty}}$ contains a certain perfect set, which means the boundary $\partial U$ has a very complicated structure. In all cases, this result leads to the disconnectivity of the Julia set $J_f$ in ${\Bbb C}$. If $U$ is a Baker domain on which $f$ is univalent, however, we shall show by giving an example that $\partial U$ can be a Jordan arc in ${\Bbb C}$, which has a rather simple structure, and, moreover, $J_f$ can be connected.

We also consider the connectivity of the set $J_f \cup \{ \infty \}$ in the Riemann sphere $\widehat{{\Bbb C}}$ and show that $J_f \cup \{ \infty \}$ is connected if and only if $f$ has no multiply-connected wandering domains.