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

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 $$\mathcal{T}:=T_1\times T_2\times \cdots \times T_H\mathrm{.}$$ We use these conditions together with Liv\v{s}ic theorem, to obtain that for $\alpha$-H"older 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 $$\underset{N}{\sup }\underset{x\in U}{\sup }\mid \sum _{j=0}^N\prod _{i=1}^H{f}_i(T_i^{j}x)\mid <\mathrm{\infty }\mathrm{.}$$