Nicolai Haydn

Hitting and return times in ergodic dynamical systems

Given an ergodic dynamical system (X, T,μ), and U C X measurable with μ(U) > 0, let μ(U)τ U (x) denote the normalized hitting time of x ∈ X to U. We prove that given a sequence (U n ) with μ(U n ) → 0, the distribution function of the normalized hitting times to U n converges weakly to some subprobability distribution F if and only if the distribution function of the normalized return time converges weakly to some distribution function F, and that in the converging case, (=) F(t)= ∫ 0 t (1-F(s))ds, t ≥ 0. This in particular characterizes asymptotics for hitting times, and shows that the asymptotics for return times is exponential if and only if the one for hitting times is also.

Equivalence of Gibbs and equilibrium states for homeomorphisms satisfying expansiveness and specification

AbstractLetM be a compact metrizable space,f: M→M a homeomorphism satisfying expansiveness and specification, andA: M → ℝ a function such that

Topological entropy of polygon exchange transformations and polygonal billiards

\left| {\sum\limits_{k = 0}^{n - 1} {[A(f^k x) - A(f^k y)]} } \right| \mathbin{\lower.3ex\hbox{

Entry and return times distribution

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Convergence of rare event point processes to the Poisson process for planar billiards

}} K(\varepsilon )< \infty

Statistical properties of intermittent maps with unbounded derivative

whenevern≧1 andx, y are (ε,n)-close (i.e.d(fkx, fky)0). Under these conditions, Bowen has shown that there is a uniqueequilibrium state ρ forA. Assuming thatK(δ)→0 when δ→0, we show that ρ is also the uniqueGibbs state forA. We further definequasi-Gibbs states and show that ρ is the uniquef-invariant quasi-Gibbs state forA.

A note on Borel-Cantelli lemmas for non-uniformly hyperbolic dynamical systems

We prove that the transfer operator for a general class of rational maps converges exponentially fast in the supremum norm and in Holder norms, for small enough Holder exponents, to its principal eigendirection.

Gibb's functionals on subshifts

We study the topological entropy of a class of transformations with mild singularities: the generalized polygon exchanges. This class contains, in particular, polygonal billiards. Our main result is a geometric estimate, from above, on the topological entropy of generalized polygon exchanges. One of the applications of our estimate is that the topological entropy of polygonal billiards is zero. This implies the subexponential growth of various geometric quantities associated with a polygon. Other applications are to the piecewise isometries in two dimensions, and to billiards in rational polyhedra.