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 , where is a slowly varying function and is determined by the speed of mixing. We strongly improve previous results where the best available rates did not exceed . To break the 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 be a dynamically coherent partially hyperbolic diffeomorphism whose center foliation has all its leaves compact. We prove that if the unstable bundle of is one-dimensional, then the volume of center leaves must be bounded in .

]]>Group actions on a Smale space and the actions induced on the -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 -algebras. In each of these cases, we discuss the preservation of properties (such as finite nuclear dimension, -stability, and classification by Elliott invariants) in the resulting crossed products.

]]>Let be a tuple of real matrices. Under certain irreducibility assumptions, we give checkable criteria for deciding whether possesses the following property: there exist two constants and such that for any and any , either or , where 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 , the absolute continuity of certain self-affine measures in and the dimensional regularity of a class of sofic affine-invariant sets in the plane.

]]>Let be a compact, metric and totally disconnected space and let be a continuous map. We relate the eigenvalues of to dynamical properties of , 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 below by the spectral radius of .

]]>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 is disjoint from all minimal systems if and only if is weakly mixing and there is some countable dense subset of such that for any minimal system , any point and any open neighbourhood of , and for any non-empty open subset , there is such that is syndetic. Some characterization for the general case is also given. By way of application we show that if a transitive system is disjoint from all minimal systems, then so are and for any . It turns out that a transitive system is disjoint from all minimal systems if and only if the hyperspace system is disjoint from all minimal systems.

]]>For the -transformation is defined by . For let be the survivor set of with hole given by

We study 2-generated subgroups such that is isomorphic to Thompson’s group , and such that the supports of and 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 diffeomorphisms on 1-manifolds, examples which do not admit faithful actions by 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 as part of a general phenomenon among subgroups of .

]]>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 -odometers give rise to counterexamples to the HK conjecture, which relates the homology of an essentially principal, minimal, ample groupoid with the K-theory of . We also show that transformation groupoids of odometers satisfy the AH conjecture.

]]>Let be a shift of finite type and its corresponding automorphism group. Associated to are certain Lyapunov exponents , which describe asymptotic behavior of the sequence of coding ranges of . We give lower bounds on in terms of the spectral radius of the corresponding action of on the dimension group associated to . We also give lower bounds on the topological entropy in terms of a distinguished part of the spectrum of the action of on the dimension group, but show that, in general, is not bounded below by the logarithm of the spectral radius of the action of on the dimension group.

]]>For a non-generic, yet dense subset of 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 perturbation theorem which allows us to interpolate between expanding Markov maps and the shift map on a finite number of symbols.

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