This paper concerns Smale diffeomorphisms of compact oriented surfaces. Relationships are found between the isotopy class of the map and the dynamics of its basic sets. The form of the dynamical properties involves restrictions on periods and reduced zeta functions.

Let ${\mathbf {G}}$ be a semisimple algebraic group over a number field K, $\mathcal {S}$ a finite set of places of K, $K_{\mathcal {S}}$ the direct product of the completions $K_{v}, v \in \mathcal {S}$ , and ${\mathcal O}$ the ring of $\mathcal {S}$ -integers of K. Let $G = {\mathbf {G}}(K_{\mathcal {S}})$ , $\Gamma = {\mathbf {G}}({\mathcal O})$ and $\pi :G \rightarrow G/\Gamma $ the quotient map. We describe the closures of the locally divergent orbits ${T\pi (g)}$ where T is a maximal $K_{\mathcal {S}}$ -split torus in G. If $\# S = 2$ then the closure $ \overline{T\pi (g)}$ is a finite union of T-orbits stratified in terms of parabolic subgroups of ${\mathbf {G}} \times {\mathbf {G}}$ and, consequently, $\overline{T\pi (g)}$ is homogeneous (i.e. $\overline{T\pi (g)}= H\pi (g)$ for a subgroup H of G) if and only if ${T\pi (g)}$ is closed. On the other hand, if $\# \mathcal {S}> 2$ and K is not a $\mathrm {CM}$ -field then $\overline {T\pi (g)}$ is homogeneous for ${\mathbf {G}} = \mathbf {SL}_{n}$ and, generally, non-homogeneous but squeezed between closed orbits of two reductive subgroups of equal semisimple K-ranks for ${\mathbf {G}} \neq \mathbf {SL}_{n}$ . As an application, we prove that $\overline {f({\mathcal O}^{n})} = K_{\mathcal {S}}$ for the class of non-rational locally K-decomposable homogeneous forms $f \in K_{\mathcal {S}}[x_1, \ldots , x_{n}]$ .

We study homeomorphisms of a Cantor set with $k$ ($k<+\infty$) minimal invariant closed (but not open) subsets; we also study crossed product C*-algebras associated to these Cantor systems and certain of their orbit-cut sub-C*-algebras. In the case where $k\geq 2$, the crossed product C*-algebra is stably finite, has stable rank 2, and has real rank 0 if in addition $(X,\unicode[STIX]{x1D70E})$ is aperiodic. The image of the index map is connected to certain directed graphs arising from the Bratteli–Vershik–Kakutani model of the Cantor system. Using this, it is shown that the ideal of the Bratteli diagram (of the Bratteli–Vershik–Kakutani model) must have at least $k$ vertices at each level, and the image of the index map must consist of infinitesimals.

We develop a thermodynamic formalism for a smooth realization of pseudo-Anosov surface homeomorphisms. In this realization, the singularities of the pseudo-Anosov map are assumed to be fixed, and the trajectories are slowed down so the differential is the identity at these points. Using Young towers, we prove existence and uniqueness of equilibrium states for geometric t-potentials. This family of equilibrium states includes a unique SRB measure and a measure of maximal entropy, the latter of which has exponential decay of correlations and the central limit theorem.

It is proved that a homeomorphism h, which conjugates a smooth unimodal map of the interval with negative Schwarzian derivative and positive Lyapunov exponent along the forward trajectory of the critical value with a tent map, and its inverse h−1 are Hölder continuous.

Let ${\mathcal{T}}$ be a locally finite tree, $\unicode[STIX]{x1D6E4}$ be a discrete subgroup of $\,\operatorname{Aut}\,({\mathcal{T}})$ and $\widetilde{F}$ be a $\unicode[STIX]{x1D6E4}$-invariant potential. Suppose that the length spectrum of $\unicode[STIX]{x1D6E4}$ is not arithmetic. In this case, we prove the exponential mixing property of the geodesic translation map $\unicode[STIX]{x1D719}:\unicode[STIX]{x1D6E4}\backslash S{\mathcal{T}}\rightarrow \unicode[STIX]{x1D6E4}\backslash S{\mathcal{T}}$ with respect to the measure $m_{\unicode[STIX]{x1D6E4},F}^{\unicode[STIX]{x1D708}^{-},\unicode[STIX]{x1D708}^{+}}$ under the assumption that $\unicode[STIX]{x1D6E4}$ is full and $(\unicode[STIX]{x1D6E4},\widetilde{F})$ has a weighted spectral gap. We also obtain the effective formula for the number of $\unicode[STIX]{x1D6E4}$-orbits with weights in a Bruhat–Tits tree ${\mathcal{T}}$ of an algebraic group.

We introduce the notion of balanced strong shift equivalence between square non-negative integer matrices, and show that two finite graphs with no sinks are one-sided eventually conjugate if and only if their adjacency matrices are conjugate to balanced strong shift equivalent matrices. Moreover, we show that such graphs are eventually conjugate if and only if one can be reached by the other via a sequence of out-splits and balanced in-splits, the latter move being a variation of the classical in-split move introduced by Williams in his study of shifts of finite type. We also relate one-sided eventual conjugacies to certain block maps on the finite paths of the graphs. These characterizations emphasize that eventual conjugacy is the one-sided analog of two-sided conjugacy.

We show that self-similar measures on $\mathbb R^d$ satisfying the weak separation condition are uniformly scaling. Our approach combines elementary ergodic theory with geometric analysis of the structure given by the weak separation condition.

We extend our previous work by proving that for translation invariant Gibbs states on ${\mathbb Z}^d$ with a translation invariant interaction potential $\Psi=(\Psi_A)$ satisfying $\sum_{A \ni 0}|A|^{-1}[\diam(A)]^d\|\Psi_A\|<\infty$ the following hold: (1) the Kolmogorov-property implies a trivial full tail and (2) the Bernoulli-property implies Følner independence. The existence of bilaterally deterministic Bernoulli Shifts tells us that neither (1) nor (2) is, in general, true for random fields without some further assumption (even when $d=1$).

It is known that volume hyperbolicity (partial hyperbolicity and uniform expansion or contraction of the volume in the extremal bundles) is a necessary condition for robust transitivity or robust chain recurrence and hence for tameness. In this paper, on any $3$-manifold we build examples of quasi-attractors which are volume hyperbolic and wild at the same time. As a main corollary, we see that, for any closed $3$-manifold $M$, the space $\text{Diff}^{1}(M)$ admits a non-empty open set where every $C^{1}$-generic diffeomorphism has no attractors or repellers. The main tool of our construction is the notion of flexible periodic points introduced in the authors’ previous paper. In order to eject the flexible points from the quasi-attractor, we control the topology of the quasi-attractor using the notion of partially hyperbolic filtrating Markov partitions, which we introduce in this paper.

We consider a locally path-connected compact metric space K with finite first Betti number $\textrm {b}_1(K)$ and a flow $(K, G)$ on K such that G is abelian and all G-invariant functions $f\,{\in}\, \textrm{C}(K)$ are constant. We prove that every equicontinuous factor of the flow $(K, G)$ is isomorphic to a flow on a compact abelian Lie group of dimension less than ${\textrm {b}_1(K)}/{\textrm {b}_0(K)}$ . For this purpose, we use and provide a new proof for Theorem 2.12 of Hauser and Jäger [Monotonicity of maximal equicontinuous factors and an application to toral flows. Proc. Amer. Math. Soc.147 (2019), 4539–4554], which states that for a flow on a locally connected compact space the quotient map onto the maximal equicontinuous factor is monotone, i.e., has connected fibers. Our alternative proof is a simple consequence of a new characterization of the monotonicity of a quotient map $p\colon K\to L$ between locally connected compact spaces K and L that we obtain by characterizing the local connectedness of K in terms of the Banach lattice $\textrm {C}(K)$ .

Suppose that a countable group $G$ admits a cusp-uniform action on a hyperbolic space $(X,d)$ such that $G$ is of divergent type. The main result of the paper is characterizing the purely exponential growth type of the orbit growth function by a condition introduced by Dal’bo, Otal and Peigné [Séries de Poincaré des groupes géométriquement finis. Israel J. Math.118(3) (2000), 109–124]. For geometrically finite Cartan–Hadamard manifolds with pinched negative curvature, this condition ensures the finiteness of Bowen–Margulis–Sullivan measures. In this case, our result recovers a theorem of Roblin (in a coarse form). Our main tool is the Patterson–Sullivan measures on the Gromov boundary of $X$, and a variant of the Sullivan shadow lemma called the partial shadow lemma. This allows us to prove that the purely exponential growth of either cones, or partial cones or horoballs is also equivalent to the Dal’bo–Otal–Peigné condition. These results are used further in a paper by the present author [W. Yang, Patterson–Sullivan measures and growth of relatively hyperbolic groups. Preprint, 2013, arXiv:1308.6326].

Let $A\in SL(2,{\Bbb Z})$ be given and $n\in{\Bbb N}$. $A$ induces a map $A_n$ on ${\Bbb Z}^2_n$ by $(x,y)\mapsto(a_{11}x+a_{12}y,a_{21}x+a_{22}y)$ mod $n$. $A_n$ can be thought of as a discrete approximation of the similarly defined map on the unit square. It is known that the $A_n$ have a surprisingly small period: there always is an $r\le3n$ such that $A^r_n$ is the identity on ${\Bbb Z}^2_n$.

One can visualize the properties of the iterates of $A_n$ by considering the sets $P,A_nP,A^2_nP,\ldots$ for a fixed subset $P$ of ${\Bbb Z}^2_n$. (Here the cat comes into play: $P$ has often been chosen to be the digitalized picture of a cat, in particular when treating the special case of the Anosov map $A={2\ 1\atopwithdelims() 1\ 1}$.) It turns out that in the sequence $(A^k_nP)_k$ mostly ‘chaotic’ pictures occur as one would expect. However, some kind of regular behaviour can also be observed for certain powers $A^k_n$ before the end of the period.

It is the aim of this paper to describe these phenomena and to explain why and how they occur. In order to quantify the chaotic behaviour of the maps on ${\Bbb Z}^2_n$ under consideration, we introduce suitable functions.

Our methods are elementary. Only basic results from the geometry of numbers will be used.

Fathi and Franks showed that a pseudo-Anosov diffeomorphism $f$ with orientable foliations and dilation coefficient $\lambda$ with no conjugates (over ${\mathbb Q}$) in the unit circle factors onto a (homologically non-trivial) invariant subset of a hyperbolic toral automorphism. After recounting this result, we show that the factor map is either almost everywhere one-to-one or almost everywhere $m$-to-one for some $m>1$ and the pseudo-Anosov map $f$ is an $m$-to-one ramified covering of another pseudo-Anosov (or Anosov) map on a surface of smaller genus. As a corollary, any pseudo-Anosov diffeomorphism with orientable foliations and hyperbolic action on the first homologies almost everywhere embeds into a hyperbolic toral automorphism.

In this paper, we consider the product space of several non-compact finite-volume hyperbolic spaces, $V_{1},V_{2},\ldots ,V_{k}$ of dimension $n$. Let $\text{T}^{1}(V_{i})$ denote the unit tangent bundle of $V_{i}$ and $g_{t}$ denote the geodesic flow on $\text{T}^{1}(V_{i})$ for each $i=1,\ldots ,k$. We define

$$\begin{eqnarray}{\mathcal{D}}_{k}:=\{(v_{1},\ldots ,v_{k})\,\in \,\text{T}^{1}(V_{1})\times \cdots \times \text{T}^{1}(V_{k})\,:\,(g_{t}(v_{1}),\ldots ,g_{t}(v_{k}))\text{ diverges as }t\rightarrow \infty \}.\end{eqnarray}$$

${\mathcal{D}}_{k}$

$k(2n-1)-((n-1)/2)$

We first improve an old result of McMahon and show that a metric minimal flow whose enveloping semigroup contains less than $2^{\mathfrak{c}}$ (where $\mathfrak{c}=2^{\aleph _{0}}$) minimal left ideals is proximal isometric (PI). Then we show the existence of various minimal PI-flows with many minimal left ideals, as follows. For the acting group $G=\text{SL}_{2}(\mathbb{R})^{\mathbb{N}}$, we construct a metric minimal PI $G$-flow with $\mathfrak{c}$ minimal left ideals. We then use this example and results established in Glasner and Weiss. [On the construction of minimal skew-products. Israel J. Math.34 (1979), 321–336] to construct a metric minimal PI cascade $(X,T)$ with $\mathfrak{c}$ minimal left ideals. We go on to construct an example of a minimal PI-flow $(Y,{\mathcal{G}})$ on a compact manifold $Y$ and a suitable path-wise connected group ${\mathcal{G}}$ of a homeomorphism of $Y$, such that the flow $(Y,{\mathcal{G}})$ is PI and has $2^{\mathfrak{c}}$ minimal left ideals. Finally, we use this latter example and a theorem of Dirbák to construct a cascade $(X,T)$ that is PI (of order three) and has $2^{\mathfrak{c}}$ minimal left ideals. Thus this final result shows that, even for cascades, the converse of the implication ‘less than $2^{\mathfrak{c}}$ minimal left ideals implies PI’, fails.

We consider a $C^\infty$ Lagrangian function $L: T^*M\rightarrow \mathbb{R}$ which is superlinear and convex in the fibers and has one antisymplectic symmetry. We prove that:

1. in every energy level strictly above the critical one, there exists a Mather set which is the union of some periodic orbits;

2. for $L$ generic, these orbits are hyperbolic;

3. on the torus, these orbits have one homoclinic orbit.

We give a definition for a rank-one homeomorphism of a zero-dimensional Polish space X. We show that if a rank-one homeomorphism of X satisfies a certain non-degeneracy condition, then it has trivial centralizer in the group of all homeomorphisms of X, i.e. it commutes only with its integral powers.

For a topological dynamical system Σ = (X, σ) where X is a compact metric space with a single homeomorphism σ, we determine the largest postliminal ideal of the transformation group C*-algebra A(Σ) as the intersection of all kernels of irreducible representations of A(Σ) induced from those recurrent points which are not periodic. The result implies characterizations of topological dynamical systems whose transformation group C*-algebras are anti-liminal and post-liminal, that is, of type 1.

We study the repetition of patches in self-affine tilings in ${\mathbb {R}}^d$ . In particular, we study the existence and non-existence of arithmetic progressions. We first show that an arithmetic condition of the expansion map for a self-affine tiling implies the non-existence of certain one-dimensional arithmetic progressions. Next, we show that the existence of full-rank infinite arithmetic progressions, pure discrete dynamical spectrum, and limit-periodicity are all equivalent for a certain class of self-affine tilings. We finish by giving a complete picture for the existence or non-existence of full-rank infinite arithmetic progressions in the self-similar tilings in ${\mathbb {R}}^d$ .