American Institute of Mathematical Sciences

Advanced
February & March  2004, 11(2&3): 261-310. doi: 10.3934/dcds.2004.11.261

Smale diffeomorphisms of surfaces: a classification algorithm

François Béguin 1,

1. 

Université Paris Sud, Département de mathématiques, 91405 Orsay, France

Received  December 2002 Revised  January 2004 Published  June 2004

We are concerned here with Smale (i.e. $C^1$-structurally stable) diffeomorphisms of compact surfaces. Bonatti and Langevin have produced some combinatorial descriptions of the dynamics of any such diffeomorphism ([2]). Actually, each diffeomorphism admits infinitely many different combinatorial descriptions. The aim of the present article is to describe an algorithm which decides whether two combinatorial descriptions correspond to the same diffeomorphism or not. This provides an algorithmic way to classify Smale diffeomorphisms of surfaces up to topological conjugacy (on canonical neighbourhoods of the basic pieces).
Keywords: axiom A diffeomorphisms, Markov partitions., Smale diffeomorphisms.
Mathematics Subject Classification: 36C15, 37D20, 37E30.
Citation: François Béguin. Smale diffeomorphisms of surfaces: a classification algorithm. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 261-310. doi: 10.3934/dcds.2004.11.261
[1]

Jean-René Chazottes, Renaud Leplaideur. Fluctuations of the nth return time for Axiom A diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 399-411. doi: 10.3934/dcds.2005.13.399
[2]

Christian Bonatti, Nancy Guelman. Axiom A diffeomorphisms derived from Anosov flows. Journal of Modern Dynamics, 2010, 4 (1) : 1-63. doi: 10.3934/jmd.2010.4.1
[3]

Renaud Leplaideur, Benoît Saussol. Large deviations for return times in non-rectangle sets for axiom a diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 327-344. doi: 10.3934/dcds.2008.22.327
[4]

João P. Almeida, Albert M. Fisher, Alberto Adrego Pinto, David A. Rand. Anosov diffeomorphisms. Conference Publications, 2013, 2013 (special) : 837-845. doi: 10.3934/proc.2013.2013.837
[5]

Kazuhiro Sakai. The oe-property of diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 581-591. doi: 10.3934/dcds.1998.4.581
[6]

Stefan Haller, Tomasz Rybicki, Josef Teichmann. Smooth perfectness for the group of diffeomorphisms. Journal of Geometric Mechanics, 2013, 5 (3) : 281-294. doi: 10.3934/jgm.2013.5.281
[7]

Enrique R. Pujals, Federico Rodriguez Hertz. Critical points for surface diffeomorphisms. Journal of Modern Dynamics, 2007, 1 (4) : 615-648. doi: 10.3934/jmd.2007.1.615
[8]

Baolin He. Entropy of diffeomorphisms of line. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4753-4766. doi: 10.3934/dcds.2017204
[9]

Masoumeh Gharaei, Ale Jan Homburg. Random interval diffeomorphisms. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 241-272. doi: 10.3934/dcdss.2017012
[10]

Robert McOwen, Peter Topalov. Groups of asymptotic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6331-6377. doi: 10.3934/dcds.2016075
[11]

Sheldon Newhouse. Distortion estimates for planar diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 345-412. doi: 10.3934/dcds.2008.22.345
[12]

Jinpeng An. Hölder stability of diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 315-329. doi: 10.3934/dcds.2009.24.315
[13]

Manfred G. Madritsch. Non-normal numbers with respect to Markov partitions. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 663-676. doi: 10.3934/dcds.2014.34.663
[14]

Michael Jakobson, Lucia D. Simonelli. Countable Markov partitions suitable for thermodynamic formalism. Journal of Modern Dynamics, 2018, 13: 199-219. doi: 10.3934/jmd.2018018
[15]

Omri M. Sarig. Bernoulli equilibrium states for surface diffeomorphisms. Journal of Modern Dynamics, 2011, 5 (3) : 593-608. doi: 10.3934/jmd.2011.5.593
[16]

Keonhee Lee, Kazumine Moriyasu, Kazuhiro Sakai. $C^1$-stable shadowing diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 683-697. doi: 10.3934/dcds.2008.22.683
[17]

Rafael Potrie. Partially hyperbolic diffeomorphisms with a trapping property. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 5037-5054. doi: 10.3934/dcds.2015.35.5037
[18]

Lorenzo J. Díaz, Todd Fisher. Symbolic extensions and partially hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1419-1441. doi: 10.3934/dcds.2011.29.1419
[19]

Ely Kerman. On primes and period growth for Hamiltonian diffeomorphisms. Journal of Modern Dynamics, 2012, 6 (1) : 41-58. doi: 10.3934/jmd.2012.6.41
[20]

Krzysztof Frączek. Polynomial growth of the derivative for diffeomorphisms on tori. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 489-516. doi: 10.3934/dcds.2004.11.489

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (17)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences