## Topological Methods in Nonlinear Analysis

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

Mahesh Nerurkar

#### 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.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 30, Number 1 (2007), 193-210.

Dates
First available in Project Euclid: 13 May 2016

https://projecteuclid.org/euclid.tmna/1463150080

Mathematical Reviews number (MathSciNet)
MR2363661

#### Citation

Nerurkar, Mahesh. Lifting ergodicity in $(G,\sigma)$-extension. Topol. Methods Nonlinear Anal. 30 (2007), no. 1, 193--210. https://projecteuclid.org/euclid.tmna/1463150080

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