Homeomorphisms group of normed vector space: Conjugacy problems and theKoopman operator

Mickaël D. Chekroun; Jean Roux

doi:10.3934/dcds.2013.33.3957

Article Contents

2013, Volume 33, Issue 9: 3957-3980. Doi: 10.3934/dcds.2013.33.3957

This issue Previous Article Liouville type theorems for poly-harmonic Navier problems Next Article The CLT for rotated ergodic sums and related processes

Homeomorphisms group of normed vector space: Conjugacy problems and the Koopman operator

Mickaël D. Chekroun^1, and
Jean Roux^2,

1.
Department of Mathematics, University of Hawaii at Manoa, Honolulu, HI 96822
2.
CERES-ERTI, École Normale Supérieure, 75005 Paris

Received: October 2011

Revised: February 2013

Published: September 2013

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Abstract

This article is concerned with conjugacy problems arising in the homeomorphisms group, Hom($F$), of unbounded subsets $F$ of normed vector spaces $E$. Given two homeomorphisms $f$ and $g$ in Hom($F$), it is shown how the existence of a conjugacy may be related to the existence of a common generalized eigenfunction of the associated Koopman operators. This common eigenfunction serves to build a topology on Hom($F$), where the conjugacy is obtained as limit of a sequence generated by the conjugacy operator, when this limit exists. The main conjugacy theorem is presented in a class of generalized Lipeomorphisms.

Keywords:

Mathematics Subject Classification: 20E45, 37C15, 39B62, 39B72, 47A75, 47B33, 54A20, 54E15, 54E25, 57S05.

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