Multifractal analysis of some multiple ergodic averages

TL;DR

This paper performs a comprehensive multifractal analysis of multiple ergodic averages on symbolic spaces, introducing a novel non-linear thermodynamic formalism and studying associated measures and invariance properties.

Contribution

It develops a new non-invariant, non-linear thermodynamic formalism and analyzes multifractal properties of multiple ergodic averages using telescopic measures.

Findings

Complete solution to multifractal analysis of multiple ergodic averages

Introduction of telescopic measures similar to Gibbs measures

Establishment of variational principle and pressure function in this context

Abstract

In this paper we study the multiple ergodic averages $$ \frac{1}{n}\sum_{k=1}^n \varphi(x_k, x_{kq}, ..., x_{k q^{\ell-1}}), \qquad (x_n) \in \Sigma_m $$ on the symbolic space $\Sigma_m ={0, 1, ..., m-1}^{\mathbb{N}^*}$ where $m\ge 2, \ell\ge 2, q\ge 2$ are integers. We give a complete solution to the problem of multifractal analysis of the limit of the above multiple ergodic averages. Actually we develop a non-invariant and non-linear version of thermodynamic formalism that is of its own interest. We study a large class of measures (called telescopic measures) and the special case of telescopic measures defined by the fixed points of some non-linear transfer operators plays a crucial role in studying our multiplicatively invariant sets. These measures share many properties with Gibbs measures in the classical thermodynamic formalism. Our work also concerns with variational principle, pressure function and Legendre transform in this new setting.