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Czech-Slovak Workshop on Discrete Dynamical Systems 2014 Abstract Topological transitivity of cylinder cocycles and discrete orbit A RTUR SIEMASZKO University of Warmia and Mazury in Olsztyn, Poland [email protected] By a cylinder transformation we mean a homeomorphism Tϕ : X × R −→ X × R (or rather a Z–action generated by it) given by the formula Tϕ (x, r) = (T x, ϕ(x) + r), where X is a compact metric space, T : X −→ X is a homeomorphism of X and ϕ : X −→ R is a continuous function (called a cocycle). Such a transformation cannot be itself minimal (Besicovitch 1951, Le Calvez and Yoccoz 1997). If a base transformation is a rotation on a compact metric group then Tϕ is topologically ergodic iff ϕ has zero mean with respect to the Haar measure and is not a coboundary, i.e. f is not of the form g − g ◦ T for any continuous g : X −→ R. Fra̧czek and Lemańczyk (2010) asked whether there exist topologically ergodic transformations having either dense or discrete orbits. We are able to construct such an example with (X, T ) being an odometer.
