January  2002, 8(1): 137-146. doi: 10.3934/dcds.2002.8.137

Dynamically defined recurrence dimension

1. 

Faculty of Mathematics and Physics, Charles University in Prague, Czech Republic

2. 

PHYMAT, Université de Toulon et du Var, Centre de Physique Théorique, Luminy, France, France

Received  August 2000 Revised  July 2001 Published  October 2001

We modify the idea of a previous article [8] and introduce polynomial and exponential dynamically defined recurrence dimensions, topological invariants which express how the Poincaré recurrence time of a set grows when the diameter of the set shrinks. We introduce also the concept of polynomial entropy which applies in the case that topological entropy is zero and complexity function is polynomial. We compare recurrence dimensions with topological and polynomial entropies, evaluate recurrence dimensions of Sturmian subshifts and show some examples with Toeplitz subshifts.
Citation: Petr Kůrka, Vincent Penné, Sandro Vaienti. Dynamically defined recurrence dimension. Discrete and Continuous Dynamical Systems, 2002, 8 (1) : 137-146. doi: 10.3934/dcds.2002.8.137
[1]

Milton Ko. Rényi entropy and recurrence. Discrete and Continuous Dynamical Systems, 2013, 33 (6) : 2403-2421. doi: 10.3934/dcds.2013.33.2403

[2]

Dou Dou. Minimal subshifts of arbitrary mean topological dimension. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1411-1424. doi: 10.3934/dcds.2017058

[3]

Chihurn Kim, Dong Han Kim. On the law of logarithm of the recurrence time. Discrete and Continuous Dynamical Systems, 2004, 10 (3) : 581-587. doi: 10.3934/dcds.2004.10.581

[4]

Jean René Chazottes, F. Durand. Local rates of Poincaré recurrence for rotations and weak mixing. Discrete and Continuous Dynamical Systems, 2005, 12 (1) : 175-183. doi: 10.3934/dcds.2005.12.175

[5]

Christopher Hoffman. Subshifts of finite type which have completely positive entropy. Discrete and Continuous Dynamical Systems, 2011, 29 (4) : 1497-1516. doi: 10.3934/dcds.2011.29.1497

[6]

Silvère Gangloff. Characterizing entropy dimensions of minimal mutidimensional subshifts of finite type. Discrete and Continuous Dynamical Systems, 2022, 42 (2) : 931-988. doi: 10.3934/dcds.2021143

[7]

Piotr Oprocha. Chain recurrence in multidimensional time discrete dynamical systems. Discrete and Continuous Dynamical Systems, 2008, 20 (4) : 1039-1056. doi: 10.3934/dcds.2008.20.1039

[8]

L'ubomír Snoha, Vladimír Špitalský. Recurrence equals uniform recurrence does not imply zero entropy for triangular maps of the square. Discrete and Continuous Dynamical Systems, 2006, 14 (4) : 821-835. doi: 10.3934/dcds.2006.14.821

[9]

Ville Salo, Ilkka Törmä. Recoding Lie algebraic subshifts. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 1005-1021. doi: 10.3934/dcds.2020307

[10]

Nasab Yassine. Quantitative recurrence of some dynamical systems preserving an infinite measure in dimension one. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 343-361. doi: 10.3934/dcds.2018017

[11]

Jean René Chazottes, E. Ugalde. Entropy estimation and fluctuations of hitting and recurrence times for Gibbsian sources. Discrete and Continuous Dynamical Systems - B, 2005, 5 (3) : 565-586. doi: 10.3934/dcdsb.2005.5.565

[12]

K. H. Kim and F. W. Roush. The Williams conjecture is false for irreducible subshifts. Electronic Research Announcements, 1997, 3: 105-109.

[13]

Serge Troubetzkoy. Recurrence in generic staircases. Discrete and Continuous Dynamical Systems, 2012, 32 (3) : 1047-1053. doi: 10.3934/dcds.2012.32.1047

[14]

Michael Blank. Recurrence for measurable semigroup actions. Discrete and Continuous Dynamical Systems, 2021, 41 (4) : 1649-1665. doi: 10.3934/dcds.2020335

[15]

Michel Benaim, Morris W. Hirsch. Chain recurrence in surface flows. Discrete and Continuous Dynamical Systems, 1995, 1 (1) : 1-16. doi: 10.3934/dcds.1995.1.1

[16]

Philippe Marie, Jérôme Rousseau. Recurrence for random dynamical systems. Discrete and Continuous Dynamical Systems, 2011, 30 (1) : 1-16. doi: 10.3934/dcds.2011.30.1

[17]

Miguel Abadi, Sandro Vaienti. Large deviations for short recurrence. Discrete and Continuous Dynamical Systems, 2008, 21 (3) : 729-747. doi: 10.3934/dcds.2008.21.729

[18]

Álvaro Bustos. Extended symmetry groups of multidimensional subshifts with hierarchical structure. Discrete and Continuous Dynamical Systems, 2020, 40 (10) : 5869-5895. doi: 10.3934/dcds.2020250

[19]

Nicolai T. A. Haydn. Phase transitions in one-dimensional subshifts. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 1965-1973. doi: 10.3934/dcds.2013.33.1965

[20]

Ronnie Pavlov, Pascal Vanier. The relationship between word complexity and computational complexity in subshifts. Discrete and Continuous Dynamical Systems, 2021, 41 (4) : 1627-1648. doi: 10.3934/dcds.2020334

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (138)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]