Download Topological transitivity of cylinder cocycles and discrete orbit

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.