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

Abstract

Given a compact and complete metric space $X$ with several continuous transformations $T_1,T_2,\dots T_H:X\to X,$ we find sufficient conditions for the existence of a point $x\in X$ such that $(x,x,\dots ,x)\in X^H$ has dense orbit for the transformation $$\mathcal{T}:=T_1\times T_2\times \cdots \times T_H.$$ We use these conditions together with Livšic theorem, to obtain that for $\alpha$-Hölder maps $f_1,f_2,\dots ,f_H:X\to \mathbb{R},$ 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 $$\underset{N}{\sup }\underset{x\in U}{\sup }\left|\sum _{j=0}^N\prod _{i=1}^H{f}_i(T_i^{j}x)\right|<\infty .$$

Numerous changes were made. There was an error in the proof of Theorem 2.10 in the previous version; this new version contains a theorem under new hypotheses that fixed it