We take the first steps towards a better understanding of continuous orbit equivalence, i.e., topological orbit equivalence with continuous cocycles. First, we characterize continuous orbit equivalence in terms of isomorphisms of $C^{\ast }$-crossed products preserving Cartan subalgebras. This is the topological analogue of the classical result by Singer and Feldman-Moore in the measurable setting. Second, we turn to continuous orbit equivalence rigidity, i.e., the question whether for certain classes of topological dynamical systems, continuous orbit equivalence implies conjugacy. We show that this is not always the case by constructing topological dynamical systems (actions of free abelian groups and also non-abelian free groups) that are continuously orbit equivalent but not conjugate. Furthermore, we prove positive rigidity results. For instance, for solvable duality groups, general topological Bernoulli actions and certain subshifts of full shifts over finite alphabets are rigid.

In a previous paper [Pavlov, A characterization of topologically completely positive entropy for shifts of finite type. Ergod. Th. & Dynam. Sys.34 (2014), 2054–2065], the author gave a characterization for when a $\mathbb{Z}^{d}$-shift of finite type has no non-trivial subshift factors with zero entropy, a property which we here call zero-dimensional topologically completely positive entropy. In this work, we study the difference between this notion and the more classical topologically completely positive entropy of Blanchard. We show that there are one-dimensional subshifts and two-dimensional shifts of finite type which have zero-dimensional topologically completely positive entropy but not topologically completely positive entropy. In addition, we show that strengthening the hypotheses of the main result of Pavlov [A characterization of topologically completely positive entropy for shifts of finite type. Ergod. Th. & Dynam. Sys.34 (2014), 2054–2065] yields a sufficient condition for a $\mathbb{Z}^{d}$-shift of finite type to have topologically completely positive entropy.

In this paper we show that any ergodic measure preserving transformation of a standard probability space which is $\text{AT}(n)$ for some positive integer $n$ has zero entropy. We show that for every positive integer $n$ any Bernoulli shift is not $\text{AT}(n)$. We also give an example of a transformation which has zero entropy but does not have property $\text{AT}(n)$ for any integer $n\geq 1$.

The canonical unitary representation of a locally compact separable group arising from an ergodic action of the group on a von Neumann algebra with separable predual preserving a faithful normal semifinite (infinite) weight is weak mixing. On the contrary, there exists a non-ergodic automorphism of a von Neumann algebra preserving a faithful normal semifinite trace such that the spectral measure and the spectral multiplicity of the induced action are respectively the Haar measure (on the unit circle) and $\infty$. Despite not even being ergodic, this automorphism has the same spectral data as that of a Bernoulli shift.

For a $\mathbb{Z}^{d}$-action $\unicode[STIX]{x1D6FC}$ by commuting homeomorphisms of a compact metric space, Lind introduced a dynamical zeta function that generalizes the dynamical zeta function of a single transformation. In this article, we investigate this function when $\unicode[STIX]{x1D6FC}$ is generated by continuous automorphisms of a compact abelian zero-dimensional group. We address Lind’s conjecture concerning the existence of a natural boundary for the zeta function and prove this for two significant classes of actions, including both zero entropy and positive entropy examples. The finer structure of the periodic point counting function is also examined and, in the zero entropy case, we show how this may be severely restricted for subgroups of prime index in $\mathbb{Z}^{d}$. We also consider a related open problem concerning the appearance of a natural boundary for the dynamical zeta function of a single automorphism, giving further weight to the Pólya–Carlson dichotomy proposed by Bell and the authors.

We study the different rates of escape of points under iteration by holomorphic self-maps of $\mathbb{C}^{\ast }=\mathbb{C}\setminus \{0\}$ for which both zero and infinity are essential singularities. Using annular covering lemmas we construct different types of orbits, including fast escaping and arbitrarily slowly escaping orbits to either zero, infinity or both. We also prove several properties about the set of fast escaping points for this class of functions. In particular, we show that there is an uncountable collection of disjoint sets of fast escaping points, each of which has $J(f)$ as its boundary.

Given a topological graph $E$, we give a complete description of tracial states on the $\text{C}^{\ast }$-algebra $\text{C}^{\ast }(E)$ which are invariant under the gauge action; there is an affine homeomorphism between the space of gauge invariant tracial states on $\text{C}^{\ast }(E)$ and Radon probability measures on the vertex space $E^{0}$ which are, in a suitable sense, invariant under the action of the edge space $E^{1}$. It is shown that if $E$ has no cycles, then every tracial state on $\text{C}^{\ast }(E)$ is gauge invariant. When $E^{0}$ is totally disconnected, the gauge invariant tracial states on $\text{C}^{\ast }(E)$ are in bijection with the states on $\text{K}_{0}(\text{C}^{\ast }(E))$.

Let $M$ be a compact manifold. We show that the identity component $\operatorname{Homeo}_{0}(M)$ of the group of self-homeomorphisms of $M$ has a well-defined quasi-isometry type, and study its large-scale geometry. Through examples, we relate this large-scale geometry to both the topology of $M$ and the dynamics of group actions on $M$. This gives a rich family of examples of non-locally compact groups to which one can apply the large-scale methods developed in previous work of the second author.

Consider $\mathscr{F}$, a non-empty set of subsets of $\mathbb{N}$. An operator $T$ on $X$ satisfies property ${\mathcal{P}}_{\mathscr{F}}$ if, for any non-empty open set $U$ in $X$, there exists $x\in X$ such that $\{n\geq 0:T^{n}x\in U\}\in \mathscr{F}$. Let $\overline{{\mathcal{B}}{\mathcal{D}}}$ be the collection of sets in $\mathbb{N}$ with positive upper Banach density. Our main result is a characterization of a sequence of operators satisfying property ${\mathcal{P}}_{\overline{{\mathcal{B}}{\mathcal{D}}}}$, for which we have used a deep result of Bergelson and McCutcheon in the vein of Szemerédi’s theorem. It turns out that operators having property ${\mathcal{P}}_{\overline{{\mathcal{B}}{\mathcal{D}}}}$ satisfy a kind of recurrence described in terms of essential idempotents of $\unicode[STIX]{x1D6FD}\mathbb{N}$. We will also discuss the case of weighted backward shifts. Finally, we obtain a characterization of reiteratively hypercyclic operators.

We prove that, for $C^{1}$-generic diffeomorphisms, if the periodic orbits contained in a homoclinic class $H(p)$ have all their Lyapunov exponents bounded away from zero, then $H(p)$ must be (uniformly) hyperbolic. This is in the spirit of the works on the stability conjecture, but with a significant difference that the homoclinic class $H(p)$ is not known isolated in advance, hence the ‘weak’ periodic orbits created by perturbations near the homoclinic class have to be guaranteed strictly inside the homoclinic class. In this sense the problem is of an ‘intrinsic’ nature, and the classical proof of the stability conjecture does not work. In particular, we construct in the proof several perturbations which are not simple applications of the connecting lemmas.

We establish self-norming central limit theorems for non-stationary time series arising as observations on sequential maps possessing an indifferent fixed point. These transformations are obtained by perturbing the slope in the Pomeau–Manneville map. We also obtain quenched central limit theorems for random compositions of these maps.

We prove the existence and uniqueness of equilibrium states for a family of partially hyperbolic systems, with respect to Hölder continuous potentials with small variation. The family comes from the projection, on the center-unstable direction, of a family of partially hyperbolic horseshoes introduced by Díaz et al [Destroying horseshoes via heterodimensional cycles: generating bifurcations inside homoclinic classes. Ergod. Th. & Dynam. Sys.29 (2009), 433–474]. For the original three-dimensional system we consider potentials with small variation, constant on local stable manifolds, obtaining existence and uniqueness of equilibrium states.

We discuss the set of subgroups of the automorphism group of a full shift and submonoids of its endomorphism monoid. We prove closure under direct products in the monoid case and free products in the group case. We also show that the automorphism group of a full shift embeds in that of an uncountable sofic shift. Some undecidability results are obtained as corollaries.

We prove that for the frame flow on a negatively curved, closed manifold of odd dimension other than 7, and a Hölder continuous potential that is constant on fibers, there is a unique equilibrium measure. Brin and Gromov’s theorem on the ergodicity of frame flows follows as a corollary. Our methods also give a corresponding result for automorphisms of the Heisenberg manifold fibering over the torus.

We consider suspension semiflows of angle-multiplying maps on the circle and study the distributions of periods of their periodic orbits. Under generic conditions on the roof function, we give an asymptotic formula on the number $\unicode[STIX]{x1D70B}(T)$ of prime periodic orbits with period $\leq T$. The error term is bounded, at least, by

$$\begin{eqnarray}\exp \biggl(\biggl(1-\frac{1}{4\lceil \unicode[STIX]{x1D712}_{\text{max}}/h_{\text{top}}\rceil }+\unicode[STIX]{x1D700}\biggr)h_{\text{top}}\cdot T\biggr)\quad \text{in the limit }T\rightarrow \infty\end{eqnarray}$$

$\unicode[STIX]{x1D700}>0$

$h_{\text{top}}$

$\unicode[STIX]{x1D712}_{\text{max}}$

Several problems in number theory when reformulated in terms of homogenous dynamics involve study of limiting distributions of translates of algebraically defined measures on orbits of reductive groups. The general non-divergence and linearization techniques, in view of Ratner’s measure classification for unipotent flows, reduce such problems to dynamical questions about linear actions of reductive groups on finite-dimensional vector spaces. This article provides general results which resolve these linear dynamical questions in terms of natural group theoretic or geometric conditions.

We associate the existence or non-existence of rotational invariant circles of an area-preserving twist map on the cylinder with a physically motivated quantity, the depinning force, which is a critical value in the depinning transition. Assume that $H:\mathbb{R}^{2}\mapsto \mathbb{R}$ is a $C^{2}$ generating function of an exact area-preserving twist map $\bar{\unicode[STIX]{x1D711}}$ and consider the tilted Frenkel–Kontorova (FK) model:

$$\begin{eqnarray}{\dot{x}}_{n}=-D_{1}H(x_{n},x_{n+1})-D_{2}H(x_{n-1},x_{n})+F,\quad n\in \mathbb{Z},\end{eqnarray}$$

$F\geq 0$

$F_{d}(\unicode[STIX]{x1D714})$

$\unicode[STIX]{x1D714}$

$\unicode[STIX]{x1D714}$

$\bar{\unicode[STIX]{x1D711}}$

$F_{d}(\unicode[STIX]{x1D714})=0$

$\unicode[STIX]{x1D714}$

$F_{d}(\unicode[STIX]{x1D714})=0$

$\bar{\unicode[STIX]{x1D711}}$

$\unicode[STIX]{x1D714}$

$F_{d}(\unicode[STIX]{x1D714})$

$\unicode[STIX]{x1D714}$