2020, 16: 1-36. doi: 10.3934/jmd.2020001

The degree of Bowen factors and injective codings of diffeomorphisms

Laboratoire de Mathématiques d'Orsay, Université Paris-Sud, F-91405 Orsay Cedex, France

Received  August 2018 Revised  July 10, 2019 Published  December 2019

We show that symbolic finite-to-one extensions of the type constructed by O. Sarig for surface diffeomorphisms induce Hölder-continuous conjugacies on large sets. We deduce this from their Bowen property. This notion, introduced in a joint work with M. Boyle, generalizes a fact first observed by R. Bowen for Markov partitions. We rely on the notion of degree from finite equivalence theory and magic word isomorphisms.

As an application, we give lower bounds on the number of periodic points first for surface diffeomorphisms (improving a result of Sarig) and for Sinaï billiards maps (building on a result of Baladi and Demers). Finally we characterize surface diffeomorphisms admitting a Hölder-continuous coding of all their aperiodic hyperbolic measures and give a slightly weaker construction preserving local compactness.

Citation: Jérôme Buzzi. The degree of Bowen factors and injective codings of diffeomorphisms. Journal of Modern Dynamics, 2020, 16: 1-36. doi: 10.3934/jmd.2020001
References:
[1]

V. Baladi and M. Demers, On the measure of maximal entropy for finite horizon Sinai billiard maps, J. Amer. Math. Soc., to appear. doi: 10.1090/jams/939.

[2]

S. Ben Ovadia, Symbolic dynamics for non-uniformly hyperbolic diffeomorphisms of compact smooth manifolds, J. Mod. Dyn., 13 (2018), 43-113.  doi: 10.3934/jmd.2018013.

[3]

R. Bowen, On Axiom A diffeomorphisms, Regional Conference Series in Mathematics, 35, American Mathematical Society, Providence, R.I., 1978.

[4]

M. Boyle and J. Buzzi, The almost Borel structure of surface diffeomorphisms, Markov shifts and their factors, J. Eur. Math. Soc. (JEMS), 19 (2017), 2739-2782.  doi: 10.4171/JEMS/727.

[5]

D. Burguet, Periodic expansiveness of smooth surface diffeomorphisms and applications, J. Eur. Math. Soc. (JEMS), to appear. doi: 10.4171/JEMS/925.

[6]

J. Buzzi, Subshifts of quasi-finite type, Invent. Math., 159 (2005), 369-406.  doi: 10.1007/s00222-004-0392-1.

[7]

J. Buzzi, S. Crovisier and O. Sarig, Measures of maximal entropy for surface diffeomorphisms, arXiv: 1811.02240, 2018.

[8]

N. Chernov and R. Markarian, Chaotic Billiards, Mathematical Surveys and Monographs, 127, American Mathematical Society, Providence, RI, 2006. doi: 10.1090/surv/127.

[9]

E. Covan and M. Paul, Endomorphisms of irreducible subshifts of finite type, Math. Systems Theory, 8 (1974/75), 167-175.  doi: 10.1007/BF01762187.

[10]

E. Covan and M. Paul, Sofic systems, Israel J. Math., 20 (1975), 165-177.  doi: 10.1007/BF02757884.

[11]

E. Covan and M. Paul, Finite procedures for sofic systems, Monatsh. Math., 83 (1977), 265-278.  doi: 10.1007/BF01387905.

[12]

D. Fried, Finitely presented dynamical systems, Ergodic Theory Dynam. Systems, 7 (1987), 489-507.  doi: 10.1017/S014338570000417X.

[13]

B. M. Gurevich and S. V. Savchenko, Thermodynamic formalism for symbolic Markov chains with a countable number of states, Russian Math. Surveys, 53 (1998), 245-344.  doi: 10.1070/rm1998v053n02ABEH000017.

[14]

G. Hedlund, Endomorphisms and automorphisms of the shift dynamical system, Math. Systems Theory, 3 (1969), 320-375.  doi: 10.1007/BF01691062.

[15]

V. Kaloshin, Generic diffeomorphisms with superexponential growth of number of periodic orbits, Comm. Math. Phys., 211 (2000), 253-271.  doi: 10.1007/s002200050811.

[16] A. Katok and B. Hasselblatt, An Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511809187.
[17]

A. S. Kechris, Classical Descriptive Set Theory, Graduate Texts in Mathematics, 156, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4190-4.

[18]

B. Kitchens, Symbolic Dynamics. One-Sided, Two-Sided and Countable State Markov Shifts, Universitext, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-642-58822-8.

[19]

Y. Lima and C. Matheus, Symbolic dynamics for non-uniformly hyperbolic surface maps with discontinuities, Ann. Sci. ïc. Norm. Supér. (4), 51 (2018), 1-38. 

[20]

Y. Lima and O. Sarig, Symbolic dynamics for three-dimensional flows with positive topological entropy, J. Eur. Math. Soc. (JEMS), 21 (2019), 199-256.  doi: 10.4171/JEMS/834.

[21] D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511626302.
[22]

A. Manning, Axiom A diffeomorphisms have rational zeta functions, Bull. London Math. Soc., 3 (1971), 215-220.  doi: 10.1112/blms/3.2.215.

[23]

S. Newhouse, Continuity properties of entropy, Ann. of Math. (2), 129 (1989), 215-235.  doi: 10.2307/1971492.

[24]

O. Sarig, Symbolic dynamics for surface diffeomorphisms with positive entropy, J. Amer. Math. Soc., 26 (2013), 341-426.  doi: 10.1090/S0894-0347-2012-00758-9.

[25]

O. Sarig, private communication, 2015.

show all references

References:
[1]

V. Baladi and M. Demers, On the measure of maximal entropy for finite horizon Sinai billiard maps, J. Amer. Math. Soc., to appear. doi: 10.1090/jams/939.

[2]

S. Ben Ovadia, Symbolic dynamics for non-uniformly hyperbolic diffeomorphisms of compact smooth manifolds, J. Mod. Dyn., 13 (2018), 43-113.  doi: 10.3934/jmd.2018013.

[3]

R. Bowen, On Axiom A diffeomorphisms, Regional Conference Series in Mathematics, 35, American Mathematical Society, Providence, R.I., 1978.

[4]

M. Boyle and J. Buzzi, The almost Borel structure of surface diffeomorphisms, Markov shifts and their factors, J. Eur. Math. Soc. (JEMS), 19 (2017), 2739-2782.  doi: 10.4171/JEMS/727.

[5]

D. Burguet, Periodic expansiveness of smooth surface diffeomorphisms and applications, J. Eur. Math. Soc. (JEMS), to appear. doi: 10.4171/JEMS/925.

[6]

J. Buzzi, Subshifts of quasi-finite type, Invent. Math., 159 (2005), 369-406.  doi: 10.1007/s00222-004-0392-1.

[7]

J. Buzzi, S. Crovisier and O. Sarig, Measures of maximal entropy for surface diffeomorphisms, arXiv: 1811.02240, 2018.

[8]

N. Chernov and R. Markarian, Chaotic Billiards, Mathematical Surveys and Monographs, 127, American Mathematical Society, Providence, RI, 2006. doi: 10.1090/surv/127.

[9]

E. Covan and M. Paul, Endomorphisms of irreducible subshifts of finite type, Math. Systems Theory, 8 (1974/75), 167-175.  doi: 10.1007/BF01762187.

[10]

E. Covan and M. Paul, Sofic systems, Israel J. Math., 20 (1975), 165-177.  doi: 10.1007/BF02757884.

[11]

E. Covan and M. Paul, Finite procedures for sofic systems, Monatsh. Math., 83 (1977), 265-278.  doi: 10.1007/BF01387905.

[12]

D. Fried, Finitely presented dynamical systems, Ergodic Theory Dynam. Systems, 7 (1987), 489-507.  doi: 10.1017/S014338570000417X.

[13]

B. M. Gurevich and S. V. Savchenko, Thermodynamic formalism for symbolic Markov chains with a countable number of states, Russian Math. Surveys, 53 (1998), 245-344.  doi: 10.1070/rm1998v053n02ABEH000017.

[14]

G. Hedlund, Endomorphisms and automorphisms of the shift dynamical system, Math. Systems Theory, 3 (1969), 320-375.  doi: 10.1007/BF01691062.

[15]

V. Kaloshin, Generic diffeomorphisms with superexponential growth of number of periodic orbits, Comm. Math. Phys., 211 (2000), 253-271.  doi: 10.1007/s002200050811.

[16] A. Katok and B. Hasselblatt, An Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511809187.
[17]

A. S. Kechris, Classical Descriptive Set Theory, Graduate Texts in Mathematics, 156, Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4612-4190-4.

[18]

B. Kitchens, Symbolic Dynamics. One-Sided, Two-Sided and Countable State Markov Shifts, Universitext, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-642-58822-8.

[19]

Y. Lima and C. Matheus, Symbolic dynamics for non-uniformly hyperbolic surface maps with discontinuities, Ann. Sci. ïc. Norm. Supér. (4), 51 (2018), 1-38. 

[20]

Y. Lima and O. Sarig, Symbolic dynamics for three-dimensional flows with positive topological entropy, J. Eur. Math. Soc. (JEMS), 21 (2019), 199-256.  doi: 10.4171/JEMS/834.

[21] D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511626302.
[22]

A. Manning, Axiom A diffeomorphisms have rational zeta functions, Bull. London Math. Soc., 3 (1971), 215-220.  doi: 10.1112/blms/3.2.215.

[23]

S. Newhouse, Continuity properties of entropy, Ann. of Math. (2), 129 (1989), 215-235.  doi: 10.2307/1971492.

[24]

O. Sarig, Symbolic dynamics for surface diffeomorphisms with positive entropy, J. Amer. Math. Soc., 26 (2013), 341-426.  doi: 10.1090/S0894-0347-2012-00758-9.

[25]

O. Sarig, private communication, 2015.

Figure 1.  The subshift of finite type in Example 4.9
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