Projective metrics and mixing properties on towers

Abstract

Abstract. We study the decay of correlations for towers. Using Birkhoff's projective metrics, we obtain a rate of mixing of the form:where α(n) goes to zero in a way related to the asymptotic mass of upper floors, f is some Lipschitz norm and g 1 is some L 1 norm. The fact that the dependence on g is given by an L 1 norm is useful to study asymptotic laws of successive entrance times.

Decay of correlations for the Rauzy-Veech-Zorich induction map on the space of interval exchange transformations and the central limit theorem for the Teichmüller flow on the moduli space of Abelian differentials

“…This result allows us to use the tower method of L.-S. Young [11] and to obtain the decay rate exp(−α √ n) for correlations of bounded Hölder functions. For bounded Lipschitz functions, one can also use the method of V. Maume-Deschamps [12] and obtain the uniform rate of decay at the rate exp(−αn 1/2− ). It is not clear to me, however, how to use either of these methods in the invertible case.…”

Equilibrium states, pressure and escape for multimodal maps with holes

“…The proof follows from the fact that for the transfer operator without the hole L φt , e −β gives an upper bound on the second largest eigenvalue as well as a bound on the essential spectral radius. This is proved in [M,Theorem 1.4 and Section 4.1]. There it is shown that in our setup (since β is very close to 0), a constructive bound on the second largest eigenvalue is given by tanh(R/2) where R = log 1+e −β 1−e −β .…”

A functional CLT for nonconventional polynomial arrays

“…for some constant n. In fact, also the exponential case is considered in [20] and it is possible also to get the same estimates with the norm · in place of the supremum norm.…”