Nonconventional results of Liv\v{s}ic and Gottschalk-Hedlund

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 Livsic theorem, to obtain that for $\alpha$-Holder 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 $$\sup_{N} \sup_{x\in U}\left| \sum_{j=0}^{N} \prod_{i=1}^H f_i (T_i^{j} x) \right| < \infty.$$