Topological transitivity of cylindrical cascades

The existence is proved of a topologically transitive (t.t.) homeomorphism $U$ of the space $W=\Phi\times Z$ of the form
$$ U(\varphi,z)=(T,\varphi,z+(\varphi))\quad(\varphi\in\Phi,z\in Z),\eqno(1), $$
where $\Phi$ is a complete separable metric space, $T$ is a t.t. homeomorphism of $\Phi$ onto itself, $Z$ is a separable banach space, andf is a continuous map: $\Phi\to Z$.
For the special case $W=S^1\times R$, $T\varphi=\varphi+\theta$ ($\theta$ is incommensurable with $2\pi$) the existence is proved of t.t. homeomorphisms (1) of two types: 1) with zero measure of the set of transitive points, 2) with zero measure of the set of intransitive points. An example is presented of a continuous function $f:S^1\to R$ for which the corresponding homeomorphism (1) is t.t. for all $\theta$ incommensurable with $2\pi$.