Examples of topologically transitive skew-products

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Let $\sigma:\Sigma\to\Sigma$ be a topologically mixing shift of finite type. If $G$ is a group,and $\beta:\Sigma\to G$ is a continuous function, denote by $\sigma_\beta$ the skew-product of $\sigma$ by$\beta$. If $G$ is $\mathbb R^n$, we show examples of continuous multiparameters families of functions $\beta$ for which the skew-products $\sigma_\beta$ are topologicallytransitive for sets of parameters of full measure. If $G$ isa connected semisimple matrix Lie group, we show examples of functions $\beta$ for which theskew-products $\sigma_beta$ are topologically transitive.