Ergodic Theory and Dynamical Systems最新文献

We prove a joint partial equidistribution result for common perpendiculars with given density on equidistributing equidistant hypersurfaces, towards a measure supported on truncated stable leaves. We recover a result of Marklof on the joint partial equidistribution of Farey fractions at a given density, and give several analogous arithmetic applications, including in Bruhat–Tits trees.

We show that the Hausdorff dimension of any slice of the graph of the Takagi function is bounded above by the Assouad dimension of the graph minus one, and that the bound is sharp. The result is deduced from a statement on more general self-affine sets, which is of independent interest. We also prove that Marstrand’s slicing theorem on the graph of the Takagi function extends to all slices if and only if the upper pointwise dimension of every projection of the length measure on the x-axis lifted to the graph is at least one.

We study bracket words, which are a far-reaching generalization of Sturmian words, along Hardy field sequences, which are a far-reaching generalization of Piatetski-Shapiro sequences $lfloor n^c rfloor $. We show that sequences thus obtained are deterministic (that is, they have subexponential subword complexity) and satisfy Sarnak’s conjecture.

We consider orthogonally invariant probability measures on $operatorname {mathrm {GL}}_n(mathbb {R})$ and compare the mean of the logs of the moduli of eigenvalues of the matrices with the Lyapunov exponents of random matrix products independently drawn with respect to the measure. We give a lower bound for the former in terms of the latter. The results are motivated by Dedieu and Shub [On random and mean exponents for unitarily invariant probability measures on $operatorname {mathrm {GL}}_n(mathbb {C})$. Astérisque 287 (2003), xvii, 1–18]. A novel feature of our treatment is the use of the theory of spherical polynomials in the proof of our main result.

In this work, we study ergodic and dynamical properties of symbolic dynamical system associated to substitutions on an infinite countable alphabet. Specifically, we consider shift dynamical systems associated to irreducible substitutions which have well-established properties in the case of finite alphabets. Based on dynamical properties of a countable integer matrix related to the substitution, we obtain results on existence and uniqueness of shift invariant measures.

We prove that crossed products of fiberwise essentially minimal zero-dimensional dynamical systems, a class that includes systems in which all orbit closures are minimal, have isomorphic K-theory if and only if the dynamical systems are strong orbit equivalent. Under the additional assumption that the dynamical systems have no periodic points, this gives a classification theorem including isomorphism of the associated crossed product $C^*$-algebras as well. We additionally explore the K-theory of such crossed products and the Bratteli diagrams associated to the dynamical systems.

We study equilibrium states for a class of non-uniformly expanding skew products, and show how a family of fiberwise transfer operators can be used to define the conditional measures along fibers of the product. We prove that the pushforward of the equilibrium state onto the base of the product is itself an equilibrium state for a Hölder potential defined via these fiberwise transfer operators.