For differentiable dynamical systems with dominated splittings, we give upper estimates on the measure-theoretic tail entropy in terms of Lyapunov exponents. As our primary application, we verify the upper semi-continuity of metric entropy in various settings with domination.

We prove the one-dimensional almost sure invariance principle with essentially optimal rates for slowly (polynomially) mixing deterministic dynamical systems, such as Pomeau–Manneville intermittent maps, with Hölder continuous observables. Our rates have form $o(n^{\unicode[STIX]{x1D6FE}}L(n))$, where $L(n)$ is a slowly varying function and $\unicode[STIX]{x1D6FE}$ is determined by the speed of mixing. We strongly improve previous results where the best available rates did not exceed $O(n^{1/4})$. To break the $O(n^{1/4})$ barrier, we represent the dynamics as a Young-tower-like Markov chain and adapt the methods of Berkes–Liu–Wu and Cuny–Dedecker–Merlevède on the Komlós–Major–Tusnády approximation for dependent processes.

Let $f:M\rightarrow M$ be a dynamically coherent partially hyperbolic diffeomorphism whose center foliation has all its leaves compact. We prove that if the unstable bundle of $f$ is one-dimensional, then the volume of center leaves must be bounded in $M$.

Group actions on a Smale space and the actions induced on the $\text{C}^{\ast }$-algebras associated to such a dynamical system are studied. We show that an effective action of a discrete group on a mixing Smale space produces a strongly outer action on the homoclinic algebra. We then show that for irreducible Smale spaces, the property of finite Rokhlin dimension passes from the induced action on the homoclinic algebra to the induced actions on the stable and unstable $\text{C}^{\ast }$-algebras. In each of these cases, we discuss the preservation of properties (such as finite nuclear dimension, ${\mathcal{Z}}$-stability, and classification by Elliott invariants) in the resulting crossed products.

Let $\mathbf{M}=(M_{1},\ldots ,M_{k})$ be a tuple of real $d\times d$ matrices. Under certain irreducibility assumptions, we give checkable criteria for deciding whether $\mathbf{M}$ possesses the following property: there exist two constants $\unicode[STIX]{x1D706}\in \mathbb{R}$ and $C>0$ such that for any $n\in \mathbb{N}$ and any $i_{1},\ldots ,i_{n}\in \{1,\ldots ,k\}$, either $M_{i_{1}}\cdots M_{i_{n}}=\mathbf{0}$ or $C^{-1}e^{\unicode[STIX]{x1D706}n}\leq \Vert M_{i_{1}}\cdots M_{i_{n}}\Vert \leq Ce^{\unicode[STIX]{x1D706}n}$, where $\Vert \cdot \Vert$ is a matrix norm. The proof is based on symbolic dynamics and the thermodynamic formalism for matrix products. As applications, we are able to check the absolute continuity of a class of overlapping self-similar measures on $\mathbb{R}$, the absolute continuity of certain self-affine measures in $\mathbb{R}^{d}$ and the dimensional regularity of a class of sofic affine-invariant sets in the plane.

Let $X$ be a compact, metric and totally disconnected space and let $f:X\rightarrow X$ be a continuous map. We relate the eigenvalues of $f_{\ast }:\check{H}_{0}(X;\mathbb{C})\rightarrow \check{H}_{0}(X;\mathbb{C})$ to dynamical properties of $f$, roughly showing that if the dynamics is complicated then every complex number of modulus different from 0, 1 is an eigenvalue. This stands in contrast with a classical inequality of Manning that bounds the entropy of $f$ below by the spectral radius of $f_{\ast }$.

We show that if a hyperbolic group acts geometrically on a CAT(0) cube complex, then the induced boundary action is hyperfinite. This means that for a cubulated hyperbolic group, the natural action on its Gromov boundary is hyperfinite, which generalizes an old result of Dougherty, Jackson and Kechris for the free group case.

In this paper we give an answer to Furstenberg’s problem on topological disjointness. Namely, we show that a transitive system $(X,T)$ is disjoint from all minimal systems if and only if $(X,T)$ is weakly mixing and there is some countable dense subset $D$ of $X$ such that for any minimal system $(Y,S)$, any point $y\in Y$ and any open neighbourhood $V$ of $y$, and for any non-empty open subset $U\subset X$, there is $x\in D\cap U$ such that $\{n\in \mathbb{Z}_{+}:T^{n}x\in U,S^{n}y\in V\}$ is syndetic. Some characterization for the general case is also given. By way of application we show that if a transitive system $(X,T)$ is disjoint from all minimal systems, then so are $(X^{n},T^{(n)})$ and $(X,T^{n})$ for any $n\in \mathbb{N}$. It turns out that a transitive system $(X,T)$ is disjoint from all minimal systems if and only if the hyperspace system $(K(X),T_{K})$ is disjoint from all minimal systems.

For $\unicode[STIX]{x1D6FD}\in (1,2]$ the $\unicode[STIX]{x1D6FD}$-transformation $T_{\unicode[STIX]{x1D6FD}}:[0,1)\rightarrow [0,1)$ is defined by $T_{\unicode[STIX]{x1D6FD}}(x)=\unicode[STIX]{x1D6FD}x\hspace{0.6em}({\rm mod}\hspace{0.2em}1)$. For $t\in [0,1)$ let $K_{\unicode[STIX]{x1D6FD}}(t)$ be the survivor set of $T_{\unicode[STIX]{x1D6FD}}$ with hole $(0,t)$ given by

$$\begin{eqnarray}K_{\unicode[STIX]{x1D6FD}}(t):=\{x\in [0,1):T_{\unicode[STIX]{x1D6FD}}^{n}(x)\not \in (0,t)\text{ for all }n\geq 0\}.\end{eqnarray}$$

$E_{\unicode[STIX]{x1D6FD}}$

$t\in [0,1)$

$t\mapsto K_{\unicode[STIX]{x1D6FD}}(t)$

$E_{\unicode[STIX]{x1D6FD}}$

$\unicode[STIX]{x1D6FD}\in (1,2)$

$\unicode[STIX]{x1D6FD}\in (1,2)$

$E_{\unicode[STIX]{x1D6FD}}$

$\unicode[STIX]{x1D6FD}\in (1,2)$

$E_{\unicode[STIX]{x1D6FD}}$

$E_{2}$

$\unicode[STIX]{x1D6FD}\in (1,2)$

$\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FD}}$

$K_{\unicode[STIX]{x1D6FD}}(t)$

$t<\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FD}}$

$\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D6FD}}\leq 1-(1/\unicode[STIX]{x1D6FD})$

$\unicode[STIX]{x1D6FD}\in (1,2)$

We study 2-generated subgroups $\langle f,g\rangle <\operatorname{Homeo}^{+}(I)$ such that $\langle f^{2},g^{2}\rangle$ is isomorphic to Thompson’s group $F$, and such that the supports of $f$ and $g$ form a chain of two intervals. We show that this class contains uncountably many isomorphism types. These include examples with non-abelian free subgroups, examples which do not admit faithful actions by $C^{2}$ diffeomorphisms on 1-manifolds, examples which do not admit faithful actions by $PL$ homeomorphisms on an interval, and examples which are not finitely presented. We thus answer questions due to Brin. We also show that many relatively uncomplicated groups of homeomorphisms can have very complicated square roots, thus establishing the behavior of square roots of $F$ as part of a general phenomenon among subgroups of $\operatorname{Homeo}^{+}(I)$.

We show that if there exists a counter example for the rational case of the Franks–Misiurewicz conjecture, then it must exhibit unbounded deviations in the complementary direction of its rotation set.

We compute the homology groups of transformation groupoids associated with odometers and show that certain $(\mathbb{Z}\rtimes \mathbb{Z}_{2})$-odometers give rise to counterexamples to the HK conjecture, which relates the homology of an essentially principal, minimal, ample groupoid $G$ with the K-theory of $C_{r}^{\ast }(G)$. We also show that transformation groupoids of odometers satisfy the AH conjecture.

Let $(X_{A},\unicode[STIX]{x1D70E}_{A})$ be a shift of finite type and $\text{Aut}(\unicode[STIX]{x1D70E}_{A})$ its corresponding automorphism group. Associated to $\unicode[STIX]{x1D719}\in \text{Aut}(\unicode[STIX]{x1D70E}_{A})$ are certain Lyapunov exponents $\unicode[STIX]{x1D6FC}^{-}(\unicode[STIX]{x1D719}),\unicode[STIX]{x1D6FC}^{+}(\unicode[STIX]{x1D719})$, which describe asymptotic behavior of the sequence of coding ranges of $\unicode[STIX]{x1D719}^{n}$. We give lower bounds on $\unicode[STIX]{x1D6FC}^{-}(\unicode[STIX]{x1D719}),\unicode[STIX]{x1D6FC}^{+}(\unicode[STIX]{x1D719})$ in terms of the spectral radius of the corresponding action of $\unicode[STIX]{x1D719}$ on the dimension group associated to $(X_{A},\unicode[STIX]{x1D70E}_{A})$. We also give lower bounds on the topological entropy $h_{\text{top}}(\unicode[STIX]{x1D719})$ in terms of a distinguished part of the spectrum of the action of $\unicode[STIX]{x1D719}$ on the dimension group, but show that, in general, $h_{\text{top}}(\unicode[STIX]{x1D719})$ is not bounded below by the logarithm of the spectral radius of the action of $\unicode[STIX]{x1D719}$ on the dimension group.

For a non-generic, yet dense subset of $C^{1}$ expanding Markov maps of the interval we prove the existence of uncountably many Lyapunov optimizing measures which are ergodic, fully supported and have positive entropy. These measures are equilibrium states for some Hölder continuous potentials. We also prove the existence of another non-generic dense subset for which the optimizing measure is unique and supported on a periodic orbit. A key ingredient is a new $C^{1}$ perturbation theorem which allows us to interpolate between expanding Markov maps and the shift map on a finite number of symbols.