Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Transformations conjugate to their
inverses have even essential values


Authors: Geoffrey Goodson and Mariusz Lemanczyk
Journal: Proc. Amer. Math. Soc. 124 (1996), 2703-2710
MSC (1991): Primary 28D05; Secondary 47A35
DOI: https://doi.org/10.1090/S0002-9939-96-03344-8
MathSciNet review: 1327016
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $T$ be an ergodic automorphism defined on a standard Borel probability space for which $T$ and $T^{-1}$ are isomorphic. We study the structure of the conjugating automorphisms and attempt to gain information about the structure of $T$. It was shown in Ergodic transformations conjugate to their inverses by involutions by Goodson et al. (Ergodic Theory and Dynamical Systems 16 (1996), 97--124) that if $T$ is ergodic having simple spectrum and isomorphic to its inverse, and if $S$ is a conjugation between $T$ and $T^{-1}$ (i.e. $S$ satisfies $TS=ST^{-1}$), then $S^{2}=I$, the identity automorphism. We give a new proof of this result which shows even more, namely that for such a conjugation $S$, the unitary operator induced by $T$ on $L^{2}(X,\mu )$ must have a multiplicity function whose essential values on the ortho-complement of the subspace $\{ f\in L^{2}(X,\mu ): f(S^{2})=f \} $ are always even. In particular, we see that $S$ can be weakly mixing, so the corresponding $T$ must have even maximal spectral multiplicity (regarding $\infty $ as an even number).


References [Enhancements On Off] (What's this?)

  • [1] A. Fathi, Le groupe de transformations de $[0,1]$ qui preservant la measure de Lebesgue est simple, Isr. J. Math. 29 (1978), 302-308. MR 58:6156
  • [2] G.R. Goodson, A. del Junco, M. Lema\'{n}czyk, D.J. Rudolph, Ergodic transformations conjugate to their inverses by involutions, Ergodic Theory and Dynamical Systems 16 (1996), 97--124.
  • [3] A. del Junco, M. Lema\'{n}czyk, Generic spectral properties of measure preserving maps and actions, Proc. A.M.S. 115 (1992), 725-736. MR 92i:28017
  • [4] A. Katok, unpublished lecture notes.
  • [5] M. Lema\'{n}czyk, Toeplitz-${Z \! \! \! Z}_{2}$ extensions, Ann. Inst. Henri Poinca\'{r}e 24 (1988), 1-43. MR 90b:28020
  • [6] J. Mathew, M.G. Nadkarni, A measure preserving transformation whose spectrum has a Lebesgue component of multiplicity two, Bull. Lond. Math. Soc. 16 (1984), 402-406. MR 86b:28017
  • [7] M. Queffelec, Substitution Dynamical Systems, Spectral Analysis: Lecture Notes in Math., vol. 1294, 1987. MR 89g:54094
  • [8] V.V. Ryzhikov, Representation of transformations preserving Lebesgue measure as a composition of periodic transformations, Mat. Zametky 38 (1985), 860-865; English transl., Math. Notes 38 (1985), 978--981. MR 87b:28018

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 28D05, 47A35

Retrieve articles in all journals with MSC (1991): 28D05, 47A35


Additional Information

Geoffrey Goodson
Affiliation: Department of Mathematics, Towson State University, Towson, Maryland 21204-7097
Email: e7m2grg@toe.towson.edu

Mariusz Lemanczyk
Affiliation: Institute of Mathematics, Nicholas Copernicus University, ul. Chopina 12/18, 87-100 Torun, Poland
Email: mlem@mat.uni.torun.pl

DOI: https://doi.org/10.1090/S0002-9939-96-03344-8
Received by editor(s): November 1, 1994
Received by editor(s) in revised form: February 27, 1995
Additional Notes: The second author was partially supported by a KBN grant.
Communicated by: Mary Rees
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society