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

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.

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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