Taiwanese Journal of Mathematics

Some Remarks on Dynamical System of Solenoids

Andrzej Biś and Wojciech Kozłowski

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Abstract

We show that a solenoid is a dynamical object and we express its complexity by a number of different entropy-like quantities in Hurley's sense. Some relations between these entropy-like quantities are presented. We adopt the theory of Carathéodory dimension structures introduced axiomatically by Pesin to a case of a solenoid. Any of the above mentioned entropy-like quantities determines a particular Carathéodory structure such that its upper capacity coincides with the considered quantity. We mimic a definition of the local measure entropy, introduced by Brin and Katok for a single map, to a case of a solenoid. Lower estimations of these quantities by corresponding local measure entropies are described.

Article information

Source
Taiwanese J. Math., Volume 22, Number 6 (2018), 1463-1478.

Dates
Received: 3 August 2017
Revised: 4 May 2018
Accepted: 20 May 2018
First available in Project Euclid: 9 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1528509852

Digital Object Identifier
doi:10.11650/tjm/180506

Mathematical Reviews number (MathSciNet)
MR3880237

Zentralblatt MATH identifier
1405.37022

Subjects
Primary: 37B45: Continua theory in dynamics 28D20: Entropy and other invariants
Secondary: 54H20: Topological dynamics [See also 28Dxx, 37Bxx] 54F45: Dimension theory [See also 55M10]

Keywords
entropy local measure entropy entropy-like quantity solenoids Carathéodory structure

Citation

Biś, Andrzej; Kozłowski, Wojciech. Some Remarks on Dynamical System of Solenoids. Taiwanese J. Math. 22 (2018), no. 6, 1463--1478. doi:10.11650/tjm/180506. https://projecteuclid.org/euclid.twjm/1528509852


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References

  • J. P. Boroński and P. Oprocha, On entropy of graph maps that give hereditarily indecomposable inverse limits, J. Dynam. Differential Equations 29 (2017), no. 2, 685–699.
  • M. Brin and A. Katok, On local entropy, in: Geometric Dynamics (Rio de Janeiro, 1981), 30–38, Lecture Notes in Math. 1007, Springer, Berlin, 1983.
  • W.-C. Cheng and S. E. Newhouse, Pre-image entropy, Ergodic Theory Dynam. Systems 25 (2005), no. 4, 1091–1113.
  • M. Hurley, On topological entropy of maps, Ergodic Theory Dynam. Systems 15 (1995), no. 3, 557–568.
  • W. T. Ingram and W. S. Mahavier, Inverse Limits: From continua to chaos, Developments in Mathematics 25, Springer, New York, 2012.
  • J. Kwapisz, Homotopy and dynamics for homeomorphisms of solenoids and Knaster continua, Fund. Math. 168 (2001), no. 3, 251–278.
  • R. Langevin and F. Przytycki, Entropie de l'image inverse d'une application, Bull. Soc. Math. France 120 (1992), no. 2, 237–250.
  • R. Langevin and P. Walczak, Entropie d'une dynamique, C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), no. 1, 141–144.
  • M. Lyubich and Y. Minsky, Laminations in holomorphic dynamics, J. Differential Geom. 47 (1997), no. 1, 17–94.
  • J.-H. Ma and Z.-Y. Wen, A Billingsley type theorem for Bowen entropy, C. R. Math. Acad. Sci. Paris 346 (2008), no. 9-10, 503–507.
  • M. C. McCord, Inverse limit sequences with covering maps, Trans. Amer. Math. Soc. 114 (1965), 197–209.
  • Z. Nitecki and F. Przytycki, Preimage entropy for mappings, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 9 (1999), no. 9, 1815–1843.
  • Y. B. Pesin, Dimension Theory in Dynamical Systems: Contemporary views and applications, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1997.
  • P. Raith, Two commuting interval maps with entropy zero whose composition has positive topological entropy, in: Iteration Theory (ECIT '02), 351–354, Grazer Math. Ber. 346, Karl-Franzens-Univ. Graz, Graz, 2004.
  • S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817.
  • D. Sullivan, Bounds, quadratic differential, and renormalization conjectures, in: American Mathematical Society centennial publications, Vol. II (Providence, RI, 1988), 417–466, Amer. Math. Soc., Providence, RI, 1992.
  • ––––, Solenoidal manifolds, J. Singul. 9 (2014), 203–205.
  • L. Vietoris, Über den höheren Zusammenhang kompakter Räume und eine Klasse von zusammenhangstreuen Abbildungen, Math. Ann. 97 (1927), no. 1, 454–472.
  • P. Walters, An Introduction to Ergodic Theory, Granduate Texts in Mathematics 79, Springer-Verlag, New York-Berlin, 1982.
  • R. F. Williams, Expanding attractors, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 169–203.