# American Institute of Mathematical Sciences

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.  Google Scholar [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.  Google Scholar [3] R. Bowen, On Axiom A diffeomorphisms, Regional Conference Series in Mathematics, 35, American Mathematical Society, Providence, R.I., 1978.  Google Scholar [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.  Google Scholar [5] D. Burguet, Periodic expansiveness of smooth surface diffeomorphisms and applications, J. Eur. Math. Soc. (JEMS), to appear. doi: 10.4171/JEMS/925.  Google Scholar [6] J. Buzzi, Subshifts of quasi-finite type, Invent. Math., 159 (2005), 369-406.  doi: 10.1007/s00222-004-0392-1.  Google Scholar [7] J. Buzzi, S. Crovisier and O. Sarig, Measures of maximal entropy for surface diffeomorphisms, arXiv: 1811.02240, 2018. Google Scholar [8] N. Chernov and R. Markarian, Chaotic Billiards, Mathematical Surveys and Monographs, 127, American Mathematical Society, Providence, RI, 2006. doi: 10.1090/surv/127.  Google Scholar [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.  Google Scholar [10] E. Covan and M. Paul, Sofic systems, Israel J. Math., 20 (1975), 165-177.  doi: 10.1007/BF02757884.  Google Scholar [11] E. Covan and M. Paul, Finite procedures for sofic systems, Monatsh. Math., 83 (1977), 265-278.  doi: 10.1007/BF01387905.  Google Scholar [12] D. Fried, Finitely presented dynamical systems, Ergodic Theory Dynam. Systems, 7 (1987), 489-507.  doi: 10.1017/S014338570000417X.  Google Scholar [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.  Google Scholar [14] G. Hedlund, Endomorphisms and automorphisms of the shift dynamical system, Math. Systems Theory, 3 (1969), 320-375.  doi: 10.1007/BF01691062.  Google Scholar [15] V. Kaloshin, Generic diffeomorphisms with superexponential growth of number of periodic orbits, Comm. Math. Phys., 211 (2000), 253-271.  doi: 10.1007/s002200050811.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [21] D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511626302.  Google Scholar [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.  Google Scholar [23] S. Newhouse, Continuity properties of entropy, Ann. of Math. (2), 129 (1989), 215-235.  doi: 10.2307/1971492.  Google Scholar [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.  Google Scholar [25] O. Sarig, private communication, 2015. Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [3] R. Bowen, On Axiom A diffeomorphisms, Regional Conference Series in Mathematics, 35, American Mathematical Society, Providence, R.I., 1978.  Google Scholar [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.  Google Scholar [5] D. Burguet, Periodic expansiveness of smooth surface diffeomorphisms and applications, J. Eur. Math. Soc. (JEMS), to appear. doi: 10.4171/JEMS/925.  Google Scholar [6] J. Buzzi, Subshifts of quasi-finite type, Invent. Math., 159 (2005), 369-406.  doi: 10.1007/s00222-004-0392-1.  Google Scholar [7] J. Buzzi, S. Crovisier and O. Sarig, Measures of maximal entropy for surface diffeomorphisms, arXiv: 1811.02240, 2018. Google Scholar [8] N. Chernov and R. Markarian, Chaotic Billiards, Mathematical Surveys and Monographs, 127, American Mathematical Society, Providence, RI, 2006. doi: 10.1090/surv/127.  Google Scholar [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.  Google Scholar [10] E. Covan and M. Paul, Sofic systems, Israel J. Math., 20 (1975), 165-177.  doi: 10.1007/BF02757884.  Google Scholar [11] E. Covan and M. Paul, Finite procedures for sofic systems, Monatsh. Math., 83 (1977), 265-278.  doi: 10.1007/BF01387905.  Google Scholar [12] D. Fried, Finitely presented dynamical systems, Ergodic Theory Dynam. Systems, 7 (1987), 489-507.  doi: 10.1017/S014338570000417X.  Google Scholar [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.  Google Scholar [14] G. Hedlund, Endomorphisms and automorphisms of the shift dynamical system, Math. Systems Theory, 3 (1969), 320-375.  doi: 10.1007/BF01691062.  Google Scholar [15] V. Kaloshin, Generic diffeomorphisms with superexponential growth of number of periodic orbits, Comm. Math. Phys., 211 (2000), 253-271.  doi: 10.1007/s002200050811.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.   Google Scholar [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.  Google Scholar [21] D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511626302.  Google Scholar [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.  Google Scholar [23] S. Newhouse, Continuity properties of entropy, Ann. of Math. (2), 129 (1989), 215-235.  doi: 10.2307/1971492.  Google Scholar [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.  Google Scholar [25] O. Sarig, private communication, 2015. Google Scholar
The subshift of finite type in Example 4.9
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