Lifting ergodicity in $(G,\sigma)$-extension
Keywords
Affine extensions, Dynamical Systems, ergodicity, cocyclesAbstract
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.Downloads
Published
2007-09-01
How to Cite
1.
NERURKAR, Mahesh. Lifting ergodicity in $(G,\sigma)$-extension. Topological Methods in Nonlinear Analysis. Online. 1 September 2007. Vol. 30, no. 1, pp. 193 - 210. [Accessed 28 March 2024].
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