Ergodic averages and free Z2ℤ^2Z2 actions (original) (raw)

Diffuse Behaviour of Ergodic Sums Over Rotations

arXiv (Cornell University), 2017

For a rotation by an irrational α on the circle and a BV function ϕ, we study the variance of the ergodic sums S L ϕ(x) := L−1 j=0 ϕ(x + jα). When α is not of constant type, we construct sequences (L N) such that, at some scale, the ergodic sums S L N ϕ satisfy an ASIP. Explicit non-degenerate examples are given, with an application to the rectangular periodic billiard in the plane. Contents 2.2. Application to step functions 12 2.3. Application to the periodic billiard in the plane 16 3. Appendix 19 3.1. CLT and ASIP for f k (n k .) 19 3.2. A remark on a result of Gaposhkin 22 References 24

On the ergodic theory of free group actions by real-analytic circle diffeomorphisms

Inventiones mathematicae, 2017

We consider finitely generated groups of real-analytic circle diffeomorphisms. We show that if such a group admits an exceptional minimal set (i.e., a minimal invariant Cantor set), then its Lebesgue measure is zero; moreover, there are only finitely many orbits of connected components of its complement. For the case of minimal actions, we show that if the underlying group is (algebraically) free, then the action is ergodic with respect to the Lebesgue measure. This provides first answers to questions due toÉ. Ghys, G. Hector and D. Sullivan.

Dispersion properties of ergodic translations

Iternational Journal of Mathematics and Mathematical Sciences, 2006

We consider irrational rotations of the circle Tx = x + α (mod 1) and study the asymptotic behavior of sums of the type S n = n−1 k=0 π k with π k = ±1, the sign being determined by the location of T k x with respect to a binary partition.

On the question of ergodicity for minimal group actions on the circle

2008

This work is devoted to the study of minimal, smooth actions of finitely generated groups on the circle. We provide a sufficient condition for such an action to be ergodic (with respect to the Lebesgue measure), and we illustrate this condition by studying two relevant examples. Under an analogous hypothesis, we also deal with the problem of the zero Lebesgue measure for exceptional minimal sets. This hypothesis leads to many other interesting conclusions, mainly concerning the stationary and conformal measures. Moreover, several questions are left open. The methods work as well for codimension-one foliations, though the results for this case are not explicitly stated.

Ergodic invariant measures for actions of SL(2, Z)

Annales De L Institut Henri Poincare-probabilites Et Statistiques, 1979

We construct uncountably many infinite a-finite continuous ergodic invariant measures for various actions of the unimodular group SL(2,Z). RESUME. Nous construisons un nombre non denombiable de mesures infinies, a-finies, continues, ergodiques et invariantes pour certaines actions du groupe SL(2, Z). This note arose from the following question of K. Schmidt (Oberwolfach June 1978). Consider the action of SL(2, Z) on lf2 as automorphisms arising from the linear action on (~2. Does there exist an infinite a-finite continuous (non-atomic) ergodic invariant measure on lf2 with respect to this action ? We show here, using the simple strategy elicited in the lemma below, that various actions of SL(2, Z) including the one above admit uncountably many infinite a-finite continuous ergodic invariant measures. A simple proof of the uniqueness of the finite invariant measure in the above case is included, and related problems are discussed. Annales de l’Institut Henri Poincare Section B Vol. XV...

Diffusive behavior of ergodic sums over rotations

Stochastics and Dynamics

For a rotation by an irrational [Formula: see text] on the circle and a BV function [Formula: see text], we study the variance of the ergodic sums [Formula: see text]. When [Formula: see text] is not of constant type, we construct sequences [Formula: see text] such that, at some scale, the ergodic sums [Formula: see text] satisfy an ASIP. Explicit non-degenerate examples are given with an application to the rectangular periodic billiard in the plane.

On some totally ergodic functions

Pacific Journal of Mathematics, 1990

We study some classes of totally ergodic functions on locally compact Abelian groups. Among other things, we establish the following result: If R is a locally compact commutative ring, 3ί is the additive group of R, χ is a continuous character of 3$ , and p is the function from 3l n (n e N) into 3% induced by a polynomial of n variables with coefficients in R, then the function χ o p either is a trigonometric polynomial on 3ί n or all of its Fourier-Bohr coefficients with respect to any Banach mean on L°°{^n) vanish.

The spherical ergodic theorem revisited

Expositiones Mathematicae, 2009

In this paper, we give a new proof of a result of R. Jones showing almost everywhere convergence of spherical means of actions of R d on L p (X)-spaces are convergent for d ≥ 3 and p > d d − 1 .