Topological Methods in Nonlinear Analysis

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

Mahesh Nerurkar

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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

Permanent link to this document
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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References

  • R. Ellis, The construction of minimal discrete flows , Amer. J. Math., LXXXVII , no. 3 (1965), 564–574 \ref\key 2
  • H. Keynes and M. Nerurkar, Ergodicity in affine skew product toral extensions , Pacific J. Math., 123 , no. 1 (1986), 115–126 \ref\key 3
  • H. Keynes and D. Newton, Minimal $(G,\sigma )$ extensions , Pacific J. Math., 77 (1978), 145–163 \ref\key 4 ––––, Ergodicity in $(G,\sigma )$ extensions , Springer–Verlag, Lecture Notes in Math., 819 , 265–290 \ref\key 5 ––––, Minimality for non-abelian $(G,\sigma )$-extensions , Springer–Verlag, Lecture Notes in Math., 668 (1977), 173–178 \ref\key 6
  • M. Nerurkar and H. Sussmann, Construction of minimal cocycles arising from specific differential equations , Israel J. Math., 100 (1997), 309–326 \ref\key 7 ––––, Construction of ergodic cocycles arising from specific differential equations , J. Modern Dynamics, 1 , no. 2 (2007), to appear