Lifting ergodicity in $(G,\sigma)$-extension

Mahesh Nerurkar

DOI: http://dx.doi.org/10.12775/TMNA.2007.025

Abstract


Given a compact dynamical system $(X,T,m)$ and a pair $(G,\sigma)$
consisting of a compact group $G$ and a continuous group automorphism
$\sigma$ of $G$, we consider the twisted skew-product transformation on
$G\times X$ given by
$$
T_\varphi (g,x) = (\sigma [(\varphi (x)g],Tx),
$$
where $\varphi \colon X\rightarrow G$ is a continuous map. If $(X,T,m)$ is
ergodic and aperiodic, we develop a new technique to show that for a
large class of groups $G$, the set of $\varphi$'s for which the map $T_\varphi$
is ergodic (with respect to the product measure $\nu\times m$, where
$\nu$ is the normalized Haar measure on $G$) is residual in the space of
continuous maps from $X$ to $G$. The class of groups for which the
result holds contains the class of all connected abelian and the
class of all connected Lie groups. For the class of non-abelian
fiber groups, this result is the only one of its kind.

Keywords


Affine extensions; Dynamical Systems; ergodicity; cocycles

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