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)

 
 

 

A note on the existence of a largest topological factor with zero entropy


Authors: M. Lemanczyk and A. Siemaszko
Journal: Proc. Amer. Math. Soc. 129 (2001), 475-482
MSC (2000): Primary 37B40
DOI: https://doi.org/10.1090/S0002-9939-00-05892-5
Published electronically: July 27, 2000
MathSciNet review: 1800236
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract:

Given a topological system $T$ on a $\sigma$-compact Hausdorff space and its factor $S$ we show the existence of a largest topological factor $\hat{S}$ containing $S$ such that for each $\hat{S}$-invariant measure $\mu$, $h_\mu(\hat{S}\vert S)=0$. When a relative variational principle holds, $h(\hat{S})=h(S)$.


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

  • 1. F. Blanchard, A disjointness theorem involving topological entropy, Bull. Soc. Math. France 121 (1993), 465-478. MR 95e:54050
  • 2. F. Blanchard, Y. Lacroix, Zero entropy factors of topological flows, Proc. Amer. Math. Soc. 119 (1993), 985-992. MR 93m:54066
  • 3. F. Blanchard, B. Host, A. Maas, S. Martinez, D. Rudolph, Entropy pairs for a measure, Erg. Th. Dyn. Syst. 15 (1995), 621-632. MR 96m:28024
  • 4. F. Blanchard, E. Glasner, B. Host, A variation on the variational principle and applications to entropy pairs, Erg. Th. Dyn. Syst. 17 (1997), 29-43. MR 98k:54073
  • 5. R. Bowen, Topological entropy for noncompact sets, Trans. Amer. Math. Soc. 184 (1973), 125-136. MR 49:3082
  • 6. T. Downarowicz, private communication.
  • 7. T. Downarowicz, Y. Lacroix, Almost 1-1 extensions of Furstenberg-Weiss type and applications to Toeplitz flows, Studia Math. 130 (1998), 149-170. MR 2000a:28014
  • 8. T. Downarowicz, J. Serafin, Topological fiber entropy and conditional variational principles in compact non-metrizable spaces, preprint.
  • 9. H. Furstenberg, B. Weiss, On almost 1-1 extensions, Isr. J. Math. 65 (1989), 311-322. MR 90g:28020
  • 10. E. Glasner, A simple characterization of the set of $\mu$-entropy pairs and applications, Isr. J. Math. 102 (1997), 13-27. MR 98k:54076
  • 11. E. Glasner, B. Weiss, Strictly ergodic uniform positive entropy entropy models, Bull. Soc. Math. France 122 (1994), 399-412. MR 95k:28035
  • 12. E. Glasner, B. Weiss, Topological entropy of extensions, Proc. of the 1993 Aleksandria Conference Ergodic Theory and its Connection with Harmonic Analysis in: London Math. Soc. Lectures Notes Ser. 205, 299-307. MR 96b:54064
  • 13. F. Ledrappier, P. Walters, A relativised variational principle for continuous transformations, J. London Math. Soc. 16 (1977), 568-577. MR 57:16540
  • 14. M. Misiurewicz, A short proof of the variational principle for $Z^n_+$ action on a compact space, Bull. Pol. Ac. Sc. 24 (1976), 1069-1075. MR 55:3220
  • 15. Y. Pesin, B. Pitskel, Topological pressure and the variational principle for noncompact sets, Functional Anal. Appl. 18 (1984), 307-318 (in Russian).
  • 16. P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, 1982. MR 84e:28017

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 37B40

Retrieve articles in all journals with MSC (2000): 37B40


Additional Information

M. Lemanczyk
Affiliation: Faculty of Mathematics and Computer Science, Nicholas Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland
Email: mlem@mat.uni.torun.pl

A. Siemaszko
Affiliation: Department of Applied Mathematics, Olsztyn University of Agriculture and Technology, Oczapowskiego 1, 10-957 Olsztyn-Kortowo, Poland
Email: artur@art.olsztyn.pl

DOI: https://doi.org/10.1090/S0002-9939-00-05892-5
Keywords: Topological entropy, relative Pinsker factor
Received by editor(s): April 22, 1999
Published electronically: July 27, 2000
Communicated by: Michael Handel
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society