American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Uniqueness results for boundary value problems arising from finite fuel and other singular and unbounded stochastic control problems
  • DCDS Home
  • This Issue
  • Next Article
    Renormalization of diophantine skew flows, with applications to the reducibility problem
June  2008, 21(2): 501-512. doi: 10.3934/dcds.2008.21.501

Estimates of the topological entropy from below for continuous self-maps on some compact manifolds

Wacław Marzantowicz 1, and  Feliks Przytycki 2,

1. 

Faculty of Mathematics and Comp. Sci., Adam Mickiewicz University of Poznań, ul. Umultowska 87, 61-614 Poznań, Poland
2. 

Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-950 Warszawa, Poland

Received  April 2007 Revised  October 2007 Published  March 2008

Extending our results of [17], we confirm that Entropy Conjecture holds for every continuous self-map of a compact $K(\pi,1)$ manifold with the fundamental group $\pi$ torsion free and virtually nilpotent, in particular for every continuous map of an infra-nilmanifold. In fact we prove a stronger version, a lower estimate of the topological entropy of a map by the logarithm of the spectral radius of exterior power of an associated "linearization matrix" with integer entries.
    From this, referring to known estimates of Mahler measure of polynomials, we deduce some absolute lower bounds for the entropy.
Keywords: infra-nilmanifold., virtually nilpotent group, topological entropy.
Mathematics Subject Classification: Primary: 37B40; Secondary: 22E25, 37D20, 57T1.
Citation: Wacław Marzantowicz, Feliks Przytycki. Estimates of the topological entropy from below for continuous self-maps on some compact manifolds. Discrete & Continuous Dynamical Systems - A, 2008, 21 (2) : 501-512. doi: 10.3934/dcds.2008.21.501
[1]

Jonas Deré. Periodic and eventually periodic points of affine infra-nilmanifold endomorphisms. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5347-5368. doi: 10.3934/dcds.2016035
[2]

John Franks, Michael Handel. Some virtually abelian subgroups of the group of analytic symplectic diffeomorphisms of a surface. Journal of Modern Dynamics, 2013, 7 (3) : 369-394. doi: 10.3934/jmd.2013.7.369
[3]

Katrin Gelfert. Lower bounds for the topological entropy. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 555-565. doi: 10.3934/dcds.2005.12.555
[4]

Jaume Llibre. Brief survey on the topological entropy. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3363-3374. doi: 10.3934/dcdsb.2015.20.3363
[5]

Marcelo Sobottka. Topological quasi-group shifts. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 77-93. doi: 10.3934/dcds.2007.17.77
[6]

Dongkui Ma, Min Wu. Topological pressure and topological entropy of a semigroup of maps. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 545-556. doi: 10.3934/dcds.2011.31.545
[7]

Piotr Oprocha, Paweł Potorski. Topological mixing, knot points and bounds of topological entropy. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3547-3564. doi: 10.3934/dcdsb.2015.20.3547
[8]

Boris Hasselblatt, Zbigniew Nitecki, James Propp. Topological entropy for nonuniformly continuous maps. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 201-213. doi: 10.3934/dcds.2008.22.201
[9]

Michał Misiurewicz. On Bowen's definition of topological entropy. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 827-833. doi: 10.3934/dcds.2004.10.827
[10]

Jan Philipp Schröder. Ergodicity and topological entropy of geodesic flows on surfaces. Journal of Modern Dynamics, 2015, 9: 147-167. doi: 10.3934/jmd.2015.9.147
[11]

Eva Glasmachers, Gerhard Knieper, Carlos Ogouyandjou, Jan Philipp Schröder. Topological entropy of minimal geodesics and volume growth on surfaces. Journal of Modern Dynamics, 2014, 8 (1) : 75-91. doi: 10.3934/jmd.2014.8.75
[12]

Dante Carrasco-Olivera, Roger Metzger Alvan, Carlos Arnoldo Morales Rojas. Topological entropy for set-valued maps. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3461-3474. doi: 10.3934/dcdsb.2015.20.3461
[13]

César J. Niche. Topological entropy of a magnetic flow and the growth of the number of trajectories. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 577-580. doi: 10.3934/dcds.2004.11.577
[14]

Yujun Ju, Dongkui Ma, Yupan Wang. Topological entropy of free semigroup actions for noncompact sets. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 995-1017. doi: 10.3934/dcds.2019041
[15]

Yun Zhao, Wen-Chiao Cheng, Chih-Chang Ho. Q-entropy for general topological dynamical systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2059-2075. doi: 10.3934/dcds.2019086
[16]

Tao Wang, Yu Huang. Weighted topological and measure-theoretic entropy. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3941-3967. doi: 10.3934/dcds.2019159
[17]

Xiaomin Zhou. Relative entropy dimension of topological dynamical systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (11) : 6631-6642. doi: 10.3934/dcds.2019288
[18]

Xiaojun Huang, Jinsong Liu, Changrong Zhu. The Katok's entropy formula for amenable group actions. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4467-4482. doi: 10.3934/dcds.2018195
[19]

Xiaojun Huang, Zhiqiang Li, Yunhua Zhou. A variational principle of topological pressure on subsets for amenable group actions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (5) : 2687-2703. doi: 10.3934/dcds.2020146
[20]

José S. Cánovas. Topological sequence entropy of $\omega$–limit sets of interval maps. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 781-786. doi: 10.3934/dcds.2001.7.781

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (26)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences