# Viana maps and limit distributions of sums of point measures

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1 DMA - Département de Mathématiques et Applications

Abstract : This thesis consists of five articles mainly devoted to problems in dynamical systems and ergodic theory. We consider non-uniformly hyperbolic two dimensional systems and limit distributions of point measures which are absolutely continuous with respect to the Lebesgue measure. Let $f {a 0}x=a 0-x^2$ be a quadratic map where the parameter $a 0\in1,2$ is chosen such that the critical point $0$ is pre-periodic but not periodic. In Papers A and B we study skew-products $\th,x\mapsto F\th,x=g\th,f {a 0}x+\al s\th$, $\th,x\in S^1\times eal$. The functions $g:S^1\to S^1$ and $s:S^1\to-1,1$ are the base dynamics and the coupling functions, respectively, and $\al$ is a small, positive constant. Such quadratic skew-products are also called Viana maps. In Papers A and B we show for several choices of the base dynamics and the coupling function that the map $F$ has two positive Lyapunov exponents and for some cases we further show that $F$ admits also an absolutely continuous invariant probability measure. In Paper C we consider certain Bernoulli convolutions. By showing that a specific transversality property is satisfied, we deduce absolute continuity of the to these Bernoulli convolutions associated distributions. In Papers D and E we consider sequences of real numbers in the unit interval and study how they are distributed. The sequences in Paper D are given by the forward iterations of a point $x\in0,1$ under a piecewise expanding map $T a:0,1\to0,1$ depending on a parameter $a$ contained in an interval $I$. Under the assumption that each $T a$ admits a unique absolutely continuous invariant probability measure $\mu a$ and that some technical conditions are satisfied, we show that the distribution of the forward orbit $T a^jx$, $j\ge1$, is described by the distribution $\mu a$ for Lebesgue almost every parameter $a\in I$. In Paper E we apply the ideas in Paper D to certain sequences which are equidistributed in the unit interval and give a geometrical proof of an old result by Koksma.

Keywords : Viana maps Non-uniformly hyperbolic systems Absolutely continuous invariant measures Skew products piecewise expanding maps of the interval Typical points Bernoulli convolutions uniformly distributed sequences

Author: ** Daniel Schnellmann - **

Source: https://hal.archives-ouvertes.fr/