We introduce the concept of Rokhlin dimension for actions of residually finite groups on $\text{C}^{\ast }$-algebras, extending previous such notions for actions of finite groups and the integers by Hirshberg, Winter and the third author. We are able to extend most of their results to a much larger class of groups: those admitting box spaces of finite asymptotic dimension. This latter condition is a refinement of finite asymptotic dimension and has not previously been considered. In a detailed study we show that finitely generated, virtually nilpotent groups have box spaces with finite asymptotic dimension, providing a large class of examples. We show that actions with finite Rokhlin dimension by groups with finite-dimensional box spaces preserve the property of having finite nuclear dimension when passing to the crossed product $\text{C}^{\ast }$-algebra. We then establish a relation between Rokhlin dimension of residually finite groups acting on compact metric spaces and amenability dimension of the action in the sense of Guentner, Willett and Yu. We show that for free actions of infinite, finitely generated, nilpotent groups on finite-dimensional spaces, both these dimensional values are finite. In particular, the associated transformation group$\text{C}^{\ast }$-algebras have finite nuclear dimension. This extends an analogous result about $\mathbb{Z}^{m}$-actions by the first author to a significantly larger class of groups, showing that a large class of crossed products by actions of such groups fall under the remit of the Elliott classification programme. We also provide results concerning the genericity of finite Rokhlin dimension, and permanence properties with respect to the absorption of a strongly self-absorbing $\text{C}^{\ast }$-algebra.

We obtain a unique, canonical one-to-one correspondence between the space of marked postcritically finite Newton maps of polynomials and the space of postcritically minimal Newton maps of entire maps that take the form $p(z)\exp (q(z))$ for $p(z)$, $q(z)$ polynomials and $\exp (z)$, the complex exponential function. This bijection preserves the dynamics and embedding of Julia sets and is induced by a surgery tool developed by Haïssinsky.

We establish a Garden of Eden theorem for expansive algebraic actions of amenable groups with the weak specification property, i.e. for any continuous equivariant map $T$ from the underlying space to itself, $T$ is pre-injective if and only if it is surjective. In particular, this applies to all expansive principal algebraic actions of amenable groups and expansive algebraic actions of $\mathbb{Z}^{d}$ with completely positive entropy.

We weaken the assumption of summable variations in a paper by Verbitskiy [On factors of $g$-measures. Indag. Math. (N.S.)22 (2011), 315–329] to a weaker condition, Berbee’s condition, in order for a one-block factor (a single-site renormalization) of the full shift space on finitely many symbols to have a $g$-measure with a continuous $g$-function. But we also prove by means of a counterexample that this condition is (within constants) optimal. The counterexample is based on the second of our main results, where we prove that there is a critical inverse temperature in a one-sided long-range Ising model which is at most eight times the critical inverse temperature for the (two-sided) Ising model with long-range interactions.

We effect the thermodynamical formalism for the non-uniformly hyperbolic $C^{\infty }$ map of the two-dimensional torus known as the Katok map [Katok. Bernoulli diffeomorphisms on surfaces. Ann. of Math. (2)110(3) 1979, 529–547]. It is a slow-down of a linear Anosov map near the origin and it is a local (but not small) perturbation. We prove the existence of equilibrium measures for any continuous potential function and obtain uniqueness of equilibrium measures associated to the geometric $t$-potential $\unicode[STIX]{x1D711}_{t}=-t\log \mid df|_{E^{u}(x)}|$ for any $t\in (t_{0},\infty )$, $t\neq 1$, where $E^{u}(x)$ denotes the unstable direction. We show that $t_{0}$ tends to $-\infty$ as the domain of the perturbation shrinks to zero. Finally, we establish exponential decay of correlations as well as the central limit theorem for the equilibrium measures associated to $\unicode[STIX]{x1D711}_{t}$ for all values of $t\in (t_{0},1)$.

In this paper, we present results for the forcing relation on the set of braid types of periodic orbits of a rotational horseshoe on the annulus. Precisely, we are concerned with a family of periodic orbits, called the Boyland family, and we prove that for each pair $(r,s)$ of rational numbers with $r<s$ in $(0,1)$, there exists a non-monotone orbit $B_{r,s}$ in this family which has pseudo-Anosov type and rotation interval $[r,s]$. Furthermore, the forcing relation among these orbits is given by the inclusion order on their rotation sets. It is also proved that the Markov partition associated to each Boyland orbit comes from a pruning map which projects to a bimodal circle map. This family also contains the Holmes orbits $H_{p/q}$, which are the largest for the forcing order among all the $(p,q)$-orbits of the rotational horseshoe.

We generalize the notion of Schmidt games to the setting of the general Caratheódory construction. The winning sets for such generalized Schmidt games usually have large corresponding Caratheódory dimensions (e.g., Hausdorff dimension and topological entropy). As an application, we show that for every $C^{1+\unicode[STIX]{x1D703}}$-partially hyperbolic diffeomorphism $f:M\rightarrow M$ satisfying certain technical conditions, the topological entropy of the set of points with non-dense forward orbits is bounded below by the unstable metric entropy (in the sense of Ledrappier–Young) of certain invariant measures. This also gives a unified proof of the fact that the topological entropy of such a set is equal to the topological entropy of $f$, when $f$ is a toral automorphism or the time-one map of a certain non-quasiunipotent homogeneous flow.

We prove that irreducible, linearly recurrent, type W interval-exchange transformations are always mild mixing. For every irreducible permutation, the set of linearly recurrent interval-exchange transformations has full Hausdorff dimension.

We investigate how spectral properties of a measure-preserving system $(X,{\mathcal{B}},\unicode[STIX]{x1D707},T)$ are reflected in the multiple ergodic averages arising from that system. For certain sequences $a:\mathbb{N}\rightarrow \mathbb{N}$, we provide natural conditions on the spectrum $\unicode[STIX]{x1D70E}(T)$ such that, for all $f_{1},\ldots ,f_{k}\in L^{\infty }$,

$$\begin{eqnarray}\lim _{N\rightarrow \infty }\frac{1}{N}\mathop{\sum }_{n=1}^{N}\mathop{\prod }_{j=1}^{k}T^{ja(n)}f_{j}=\lim _{N\rightarrow \infty }\frac{1}{N}\mathop{\sum }_{n=1}^{N}\mathop{\prod }_{j=1}^{k}T^{jn}f_{j}\end{eqnarray}$$

$L^{2}$

$a(n)=qn+r$

$a(n)=\lfloor \unicode[STIX]{x1D703}n+\unicode[STIX]{x1D6FE}\rfloor$

$a(n)=q_{n}$

$a(n)=p_{n}$

We obtain a spectral gap characterization of strongly ergodic equivalence relations on standard measure spaces. We use our spectral gap criterion to prove that a large class of skew-product equivalence relations arising from measurable $1$-cocycles with values in locally compact abelian groups are strongly ergodic. By analogy with the work of Connes on full factors, we introduce the Sd and $\unicode[STIX]{x1D70F}$ invariants for type $\text{III}$ strongly ergodic equivalence relations. As a corollary to our main results, we show that for any type $\text{III}_{1}$ ergodic equivalence relation ${\mathcal{R}}$, the Maharam extension $\text{c}({\mathcal{R}})$ is strongly ergodic if and only if ${\mathcal{R}}$ is strongly ergodic and the invariant $\unicode[STIX]{x1D70F}({\mathcal{R}})$ is the usual topology on $\mathbb{R}$. We also obtain a structure theorem for almost periodic strongly ergodic equivalence relations analogous to Connes’ structure theorem for almost periodic full factors. Finally, we prove that for arbitrary strongly ergodic free actions of bi-exact groups (e.g. hyperbolic groups), the Sd and $\unicode[STIX]{x1D70F}$ invariants of the orbit equivalence relation and of the associated group measure space von Neumann factor coincide.

We prove that, for almost all irrational $\unicode[STIX]{x1D70C}\in (0,1)$, the Hausdorff dimension of the invariant measure of a $C^{2+\unicode[STIX]{x1D6FC}}$-smooth $(\unicode[STIX]{x1D6FC}\in (0,1))$ circle diffeomorphism with a break of size $c\in \mathbb{R}_{+}\backslash \{1\}$, with rotation number $\unicode[STIX]{x1D70C}$, is zero. This result cannot be extended to all irrational rotation numbers.

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].

We consider homeomorphisms $f,h$ generating a faithful $\mathit{BS}(1,n)$-action on a closed surface $S$, that is, $hfh^{-1}=f^{n}$ for some $n\geq 2$. According to Guelman and Liousse [Actions of Baumslag–Solitar groups on surfaces. Discrete Contin. Dyn. Syst. A 5 (2013), 1945–1964], after replacing $f$ by a suitable iterate if necessary, we can assume that there exists a minimal set $\unicode[STIX]{x1D6EC}$ of the action, included in $\text{Fix}(f)$. Here, we suppose that $f$ and $h$ are $C^{1}$ in a neighborhood of $\unicode[STIX]{x1D6EC}$ and any point $x\in \unicode[STIX]{x1D6EC}$ admits an $h$-unstable manifold $W^{u}(x)$. Using Bonatti’s techniques, we prove that either there exists an integer $N$ such that $W^{u}(x)$ is included in $\text{Fix}(f^{N})$ or there is a lower bound for the norm of the differential of $h$ depending only on $n$ and the Riemannian metric on $S$. Combining the last statement with a result of Alonso, Guelman and Xavier [Actions of solvable Baumslag–Solitar groups on surfaces with (pseudo)-Anosov elements. Discrete Contin. Dyn. Syst. to appear], we show that any faithful action of $\mathit{BS}(1,n)$ on $S$ with $h$ a pseudo-Anosov homeomorphism has a finite orbit containing singularities of $h$; moreover, if $f$ is isotopic to the identity, it is entirely contained in the singular set of $h$. As a consequence, there is no faithful $C^{1}$-action of $\mathit{BS}(1,n)$ on the torus with $h$ Anosov.

The main purpose of this paper is to study north–south Smale solenoid diffeomorphisms on $3$-manifolds by using affine Hirsch foliations. A north–south Smale solenoid diffeomorphism $f$ on a closed $3$-manifold $M$ is a diffeomorphism whose non-wandering set is composed of a Smale solenoid attractor $\unicode[STIX]{x1D6EC}_{a}$ and a Smale solenoid repeller $\unicode[STIX]{x1D6EC}_{r}$. The key observation is that a north–south Smale solenoid diffeomorphism $f$ automatically induces two non-isotopically leaf-conjugate affine Hirsch foliations ${\mathcal{H}}^{s}$ and ${\mathcal{H}}^{u}$ on the orbit space of the wandering set of $f$ (abbreviated to the wandering orbit space of $f$) by the stable and unstable manifolds of $\unicode[STIX]{x1D6EC}_{a}$ and $\unicode[STIX]{x1D6EC}_{r}$, respectively. Under this viewpoint, we build some close relationships between north–south Smale solenoid diffeomorphisms and Hirsch manifolds (the closed $3$-manifolds admitting two non-isotopically leaf-conjugate affine Hirsch foliations).

• ∙ On the one hand, the union of the wandering orbit spaces is nearly in one-to-one correspondence with the union of Hirsch manifolds.

• ∙ On the other hand, surprisingly, the topology of the wandering orbit space (Hirsch manifold) is nearly a complete invariant of north–south Smale solenoid diffeomorphisms up to semi-global conjugacy.

Moreover, as applications, we consider several more concrete questions. For instance, we prove that every diffeomorphism in many semi-global conjugacy classes of north–south Smale solenoid diffeomorphisms are not structurally stable.

Consider a countable amenable group acting by homeomorphisms on a compact metrizable space. Chung and Li asked if expansiveness and positive entropy of the action imply existence of an off-diagonal asymptotic pair. For algebraic actions of polycyclic-by-finite groups, Chung and Li proved that they do. We provide examples showing that Chung and Li’s result is near-optimal in the sense that the conclusion fails for some non-algebraic action generated by a single homeomorphism, and for some algebraic actions of non-finitely generated abelian groups. On the other hand, we prove that every expansive action of an amenable group with positive entropy that has the pseudo-orbit tracing property must admit off-diagonal asymptotic pairs. Using Chung and Li’s algebraic characterization of expansiveness, we prove the pseudo-orbit tracing property for a class of expansive algebraic actions. This class includes every expansive principal algebraic action of an arbitrary countable group.

We establish a dynamical version of the Kuratowski–Mycielski theorem on the existence of ‘large’ invariant dependent sets. We apply this result to the study of invariant chaotic sets in topological dynamical systems, simplify many known results on this topic and also obtain some new results.