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Given a compact and complete metric space $X$ with several continuoustransformations $T_1, T_2, \ldots T_H: X \to X,$ we find sufficient conditionsfor the existence of a point $x\in X$ such that $(x,x,\ldots,x)\in X^H$ hasdense orbit for the transformation $$\mathcal T:=T_1\timesT_2\times\cdots\times T_H.$$  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 $\mathcal T$ isequivalent to the existence of a non-empty open subset $U \subset X$ such that  $$\sup_{N} \sup_{x\in U}\left| \sum_{j=0}^{N} \prod_{i=1}^H f_i (T_i^{j} x)\right| < \infty.$$

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