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Continuum-wise expansiveness and specification for set-valued functions and topological entropy

Authors: Welington Cordeiro and Maria José Pacífico
Journal: Proc. Amer. Math. Soc. 144 (2016), 4261-4271
MSC (2010): Primary 37A35, 37B40, 37D45, 54C60
Published electronically: June 17, 2016
MathSciNet review: 3531177
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Abstract: We define the concept of continuum-wise expansiveness for set-valued functions. We prove that for a compact metric space $ X$ with positive topological dimension, if a set-valued function is $ cw$-expansive, then its topological entropy is positive. We also introduce the notion of point-wise specification property for set-valued functions and prove that set-valued functions with this property have positive entropy.

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

  • [ACP] A. Arbieto, W. Cordeiro, and M. J. Pacifico, cw-expansivity and entropy for flows, arXiv:1507.01198.
  • [AKM] R. L. Adler, A. G. Konheim, and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114 (1965), 309-319. MR 0175106
  • [B1] Rufus Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc. 153 (1971), 401-414. MR 0274707
  • [B2] Rufus Bowen, Periodic orbits for hyperbolic flows, Amer. J. Math. 94 (1972), 1-30. MR 0298700
  • [BW] Rufus Bowen and Peter Walters, Expansive one-parameter flows, J. Differential Equations 12 (1972), 180-193. MR 0341451
  • [CMM] Dante Carrasco-Olivera, Roger Metzger Alvan, and Carlos Arnoldo Morales Rojas, Topological entropy for set-valued maps, Discrete Contin. Dyn. Syst. Ser. B 20 (2015), no. 10, 3461-3474. MR 3411534,
  • [Fa] Albert Fathi, Expansiveness, hyperbolicity and Hausdorff dimension, Comm. Math. Phys. 126 (1989), no. 2, 249-262. MR 1027497
  • [CDY] Welington Cordeiro, Manfred Denker, and Michiko Yuri, A note on specification for iterated function systems, Discrete Contin. Dyn. Syst. Ser. B 20 (2015), no. 10, 3475-3485. MR 3411535,
  • [Do] Tomasz Downarowicz, Entropy in dynamical systems, New Mathematical Monographs, vol. 18, Cambridge University Press, Cambridge, 2011. MR 2809170
  • [Gi] Basil E. Gillam, A new set of postulates for euclidean geometry, Revista Ci., Lima 42 (1940), 869-899. MR 0004473
  • [K1] Hisao Kato, Continuum-wise expansive homeomorphisms, Canad. J. Math. 45 (1993), no. 3, 576-598. MR 1222517,
  • [KT] J. M. Kelly and T. Tennant, Topological entropy of set-valued functions, arXiv: 1509.08413v1 (2015).
  • [KS] H. B. Keynes and M. Sears, Real-expansive flows and topological dimension, Ergodic Theory Dynamical Systems 1 (1981), no. 2, 179-195. MR 661818
  • [Ma] Ricardo Mañé, Expansive homeomorphisms and topological dimension, Trans. Amer. Math. Soc. 252 (1979), 313-319. MR 534124,
  • [RT] B. E. Raines and T. Tennant, The specification property on a set-valued map an its inverse limit, arXiv: 1509.08415v1 (2015).
  • [Re] William L. Reddy, Pointwise expansion homeomorphisms, J. London Math. Soc. (2) 2 (1970), 232-236. MR 0258009
  • [Sc] S. Schwartzman, On transformations groups, Dissertation, Yale University, 1952.
  • [Utz] W. R. Utz, Unstable homeomorphisms, Proc. Amer. Math. Soc. 1 (1950), 769-774. MR 0038022
  • [Za] Alexander J. Zaslavski, Convergence of trajectories of discrete dispersive dynamical systems, Commun. Math. Anal. 4 (2008), no. 1, 10-19. MR 2365919

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Additional Information

Welington Cordeiro
Affiliation: Department of Mathematics, The Pennsylvania State University, State College, Pennsylvania 16802

Maria José Pacífico
Affiliation: Instituto de Matemática, Universidade Federal do Rio de Janeiro, C. P. 68.530 CEP 21.945-970, Rio de Janeiro, RJ, Brazil

Keywords: Continuum-wise expansiveness, set-valued functions, entropy, specification property
Received by editor(s): October 26, 2015
Published electronically: June 17, 2016
Additional Notes: The first author was partially supported by CNPq
The second author was partially supported by CNPq, PRONEX-Dyn.Syst., FAPERJ
Communicated by: Nimish A. Shah
Article copyright: © Copyright 2016 American Mathematical Society

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