$\Delta$-transitivity for several transformations and an application to the coboundary problem

TL;DR

This paper establishes conditions for the existence of points with dense orbits under multiple transformations and applies Livšic's theorem to characterize when products of functions are smooth coboundaries in this setting.

Contribution

It introduces sufficient conditions for dense orbits in multi-transformation systems and links coboundary properties to bounded sums over these orbits.

Findings

Conditions for dense orbit existence under multiple transformations.

Characterization of smooth coboundaries via bounded sums over orbits.

Application of Livšic's theorem to product functions in dynamical systems.

Abstract

Given a compact and complete metric space $X$ with several continuous transformations $T_1, T_2, \ldots T_H: X \to X,$ we find sufficient conditions for the existence of a point $x\in X$ such that $(x,x,\ldots,x)\in X^H$ has dense orbit for the transformation $$\mathcal T:=T_1\times T_2\times\cdots\times T_H.$$ We use these conditions together with Liv\v{s}ic theorem, to obtain that for $\alpha$-H\"older maps $f_1,f_2,\ldots,f_H: X\to \mathbb{R},$ the product $\prod_{i=1}^H f_i(x_i)$ is a smooth coboundary with respect to $\mathcal T$ is equivalent to the existence of a non-empty open subset $U \subset X$ such that $$\sup_{N} \sup_{x\in U}\left| \sum_{j=0}^{N} \prod_{i=1}^H f_i (T_i^{j} x) \right| < \infty.$$