37 research outputs found
Uniqueness of equilibrium measures for countable Markov shifts and multidimensional piecewise expanding maps
We prove that potentials with summable variations on topologically transitive countable Markov shifts have at most one equilibrium measure. We apply this to multidimensional piecewise expanding maps using their Markov diagrams
Ergodic properties of equilibrium measures for smooth three dimensional flows
Let be a smooth flow with positive speed and positive topological
entropy on a compact smooth three dimensional manifold, and let be an
ergodic measure of maximal entropy. We show that either is Bernoulli,
or is isomorphic to the product of a Bernoulli flow and a rotational
flow. Applications are given to Reeb flows.Comment: 32 pages, 1 figure, a section on equilibrium measures for multiples
of the geometric potential has been added, to appear in Commentarii
Mathematici Helvetic
No temporal distributional limit theorem for a.e. irrational translation
Bromberg and Ulcigrai constructed piecewise smooth functions f on the torus
such that the set of angles alpha for which the Birkhoff sums of f with respect
to the irrational translation by alpha satisfies a temporal distributional
limit theorem along the orbit of a.e. x has Hausdorff dimension one. We show
that the Lebesgue measure of this set of angles is equal to zero
Local limit theorems for inhomogeneous Markov chains
We prove the Local Limit Theorems for bounded additive functionals of
uniformly elliptic inhomogeneous Markov arrays. As an application we obtain the
precise asymptotics in the large deviation regime for bounded additive
functionals of uniformly elliptic Markov chains. The proofs rely on new
reduction theorems for Markov arrays.Comment: 212 page
Continuity properties of Lyapunov exponents for surface diffeomorphisms
We study the entropy and Lyapunov exponents of invariant measures for
smooth surface diffeomorphisms , as functions of . The main result
is an inequality relating the discontinuities of these functions. One
consequence is that for a surface diffeomorphisms, on any set of
ergodic measures with entropy bounded away from zero, continuity of the entropy
implies continuity of the exponents. Another consequence is the upper
semi-continuity of the Hausdorff dimension on the set of ergodic invariant
measures with entropy bounded away from zero. We also obtain a new criterion
for the existence of SRB measures with positive entropy