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April  2020, 40(4): 2285-2313. doi: 10.3934/dcds.2020114

A forward Ergodic Closing Lemma and the Entropy Conjecture for nonsingular endomorphisms away from tangencies

Shuhei Hayashi

Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Tokyo, Japan

Received  June 2019 Revised  November 2019 Published  January 2020

We prove a forward Ergodic Closing Lemma for nonsingular $ C^1 $ endomorphisms, claiming that the set of eventually strongly closable points is a total probability set. The "forward" means that the closing perturbation is involved along a finite part of the forward orbit of a point in a total probability set, which is the same perturbation as in Mañé's Ergodic Closing Lemma for $ C^1 $ diffeomorphisms. As an application, Shub's Entropy Conjecture for nonsingular $ C^1 $ endomorphisms away from homoclinic tangencies is proved, extending the result for $ C^1 $ diffeomorphisms by Liao, Viana and Yang.

Keywords: Ergodic closing lemma, entropy conjecture, upper semicontinuity of entropy, entropy expansiveness, homoclinic tangencies.
Mathematics Subject Classification: 37C05, 37D30, 37B40, 37G25.
Citation: Shuhei Hayashi. A forward Ergodic Closing Lemma and the Entropy Conjecture for nonsingular endomorphisms away from tangencies. Discrete & Continuous Dynamical Systems, 2020, 40 (4) : 2285-2313. doi: 10.3934/dcds.2020114
References:
[1]

C. Bonatti, L. J. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective, Encyclopaedia of Mathematical Sciences, 102. Mathematical Physics, Ⅲ. Springer-Verlag, Berlin, 2005.  Google Scholar

[2]

R. Bowen, Entropy expansive maps, Trans. Amer. Math. Soc., 164 (1972), 323-331.  doi: 10.1090/S0002-9947-1972-0285689-X.  Google Scholar

[3]

Y. T. Cao, D. W. Yang and Y. L. Zang, The entropy conjecture for dominated splitting with multi 1D centers via upper semi-continuity of the metric entropy, Nonlineality, 30 (2017), 3076-3087.  doi: 10.1088/1361-6544/aa773c.  Google Scholar

[4]

Y. L. Cao and D. W. Yang, On Pesin's entropy formula for dominated splittings without mixed behavior, J. Diff. Equ., 261 (2016), 3964-3986.  doi: 10.1016/j.jde.2016.06.012.  Google Scholar

[5]

A. Castro, The ergodic closing lemma for nonsingular endomorphisms, preprint, (2009), arXiv: 0906.2031v2. Google Scholar

[6]

S. Crovisier, Partial hyperbolicity far from homoclinic bifurcations, Advances in Math., 226 (2011), 673-726.  doi: 10.1016/j.aim.2010.07.013.  Google Scholar

[7]

J. Franks, Necessary conditions for stability of diffeomorphisms, Trans. Amer. Math. Soc., 158 (1971), 301-308.  doi: 10.1090/S0002-9947-1971-0283812-3.  Google Scholar

[8]

J. Franks and M. Misiurewicz, Topological methods in dynamics, Handbook of dynamical systems, North-Holland, Amsterdam, 1A (2002), 547-598.  doi: 10.1016/S1874-575X(02)80009-1.  Google Scholar

[9]

S. Hayashi, An extension of the ergodic closing lemma, Ergod. Th. Dynam. Sys., 30 (2010), 773-808.  doi: 10.1017/S0143385709000273.  Google Scholar

[10]

M. W. Hirsch, J. Palis, C. C. Pugh and M. Shub, Neighborhoods of hyperbolic sets, Inventiones Math., 9 (1969/70), 121-134.  doi: 10.1007/BF01404552.  Google Scholar

[11]

M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, Vol. 583. Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[12]

G. Liao, M. Viana and J. G. Yang, The entropy conjecture for diffeomorphisms away from tangencies, J. Eur. Math. Soc., 15 (2013), 2034-2060.  doi: 10.4171/JEMS/413.  Google Scholar

[13]

P. D. Liu and K. N. Lu, A note on partially hyperbolic attractors: Entropy conjecture and SRB measures, Discrete Contin. Dyn. Syst., 35 (2015), 341-352.  doi: 10.3934/dcds.2015.35.341.  Google Scholar

[14]

R. Mañé, An ergodic closing lemma, Ann. of Math., 116 (1982), 503-540.  doi: 10.2307/2007021.  Google Scholar

[15]

R. Mañé, Ergodic Theory and Differentiable Dynamics, Ergebnisse der Mathematik und Ihrer Grenzgebiete, 8. Springer-Verlag, Berlin, 1987. doi: 10.1007/978-3-642-70335-5.  Google Scholar

[16]

A. Manning, Topological entropy and the first homology group, Dynamical Systems - Warwick 1974, Lecture Notes in Math., Springer, Berlin, 468 (1975), 185-190.   Google Scholar

[17]

M. Misiurewicz, Diffeomorphisms without any measure of maximal entropy, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 21 (1973), 903-910.   Google Scholar

[18]

M. Misiurewicz, Topological conditional entropy, Studia Math., 55 (1976), 175-200.  doi: 10.4064/sm-55-2-175-200.  Google Scholar

[19]

K. Moriyasu, The ergodic closing lemma for C1 regular maps, Tokyo J. Math., 15 (1992), 172-183.  doi: 10.3836/tjm/1270130259.  Google Scholar

[20]

V. Pliss, A hypothesis due to Smale, Diff. Eq., 8 (1972), 203-214.   Google Scholar

[21]

C. C. Pugh, The closing lemma, Amer. J. Math., 89 (1967), 956-1009.  doi: 10.2307/2373413.  Google Scholar

[22]

C. C. Pugh and C. Robinson, The C1 closing lemma, including Hamiltonians, Ergod Th. Dynam. Sys., 3 (1983), 261-313.  doi: 10.1017/S0143385700001978.  Google Scholar

[23]

M. Qian and Z. S. Zhang, Ergodic theory for axiom A endomorphisms, Ergod. Th. Dynam. Sys., 15 (1995), 161-174.  doi: 10.1017/S0143385700008294.  Google Scholar

[24]

D. Ruelle and D. Sullivan, Currents, flows and diffeomorphisms, Topology, 14 (1975), 319-327.  doi: 10.1016/0040-9383(75)90016-6.  Google Scholar

[25]

R. Saghin and Z. H. Xia, The entropy conjecture for partially hyperbolic diffeomorphisms with 1-D center, Topology Appl., 157 (2010), 29-34.  doi: 10.1016/j.topol.2009.04.053.  Google Scholar

[26]

M. Shub, Dynamical systems, filtrations and entropy, Bull. Amer. Math. Soc., 80 (1974), 27-41.  doi: 10.1090/S0002-9904-1974-13344-6.  Google Scholar

[27]

M. Shub and R. F. Williams, Entropy and stability, Topology, 14 (1975), 329-338.  doi: 10.1016/0040-9383(75)90017-8.  Google Scholar

[28]

M. Urbanski and C. Wolf, SRB measures for Axiom A endomorphisms, Math. Res. Lett., 11 (2004), 785-797.  doi: 10.4310/MRL.2004.v11.n6.a6.  Google Scholar

[29]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79. Springer-Verlag, New York-Berlin, 1982.  Google Scholar

[30]

L. Wen, The C1 closing lemma for non-singular endomorphisms, Ergod. Th. Dynam. Sys., 11 (1991), 393-412.  doi: 10.1017/S0143385700006210.  Google Scholar

[31]

L. Wen, Generic diffeomorphisms away from homoclinic tangencies and heterodimensional cycles, Bull. Braz. Math. Soc. (N.S), 35 (2004), 419-452.  doi: 10.1007/s00574-004-0023-x.  Google Scholar

[32]

Y. Yomdin, Volume growth and entropy, Israel J. Math., 57 (1987), 285-300.  doi: 10.1007/BF02766215.  Google Scholar

show all references

References:
[1]

C. Bonatti, L. J. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity. A Global Geometric and Probabilistic Perspective, Encyclopaedia of Mathematical Sciences, 102. Mathematical Physics, Ⅲ. Springer-Verlag, Berlin, 2005.  Google Scholar

[2]

R. Bowen, Entropy expansive maps, Trans. Amer. Math. Soc., 164 (1972), 323-331.  doi: 10.1090/S0002-9947-1972-0285689-X.  Google Scholar

[3]

Y. T. Cao, D. W. Yang and Y. L. Zang, The entropy conjecture for dominated splitting with multi 1D centers via upper semi-continuity of the metric entropy, Nonlineality, 30 (2017), 3076-3087.  doi: 10.1088/1361-6544/aa773c.  Google Scholar

[4]

Y. L. Cao and D. W. Yang, On Pesin's entropy formula for dominated splittings without mixed behavior, J. Diff. Equ., 261 (2016), 3964-3986.  doi: 10.1016/j.jde.2016.06.012.  Google Scholar

[5]

A. Castro, The ergodic closing lemma for nonsingular endomorphisms, preprint, (2009), arXiv: 0906.2031v2. Google Scholar

[6]

S. Crovisier, Partial hyperbolicity far from homoclinic bifurcations, Advances in Math., 226 (2011), 673-726.  doi: 10.1016/j.aim.2010.07.013.  Google Scholar

[7]

J. Franks, Necessary conditions for stability of diffeomorphisms, Trans. Amer. Math. Soc., 158 (1971), 301-308.  doi: 10.1090/S0002-9947-1971-0283812-3.  Google Scholar

[8]

J. Franks and M. Misiurewicz, Topological methods in dynamics, Handbook of dynamical systems, North-Holland, Amsterdam, 1A (2002), 547-598.  doi: 10.1016/S1874-575X(02)80009-1.  Google Scholar

[9]

S. Hayashi, An extension of the ergodic closing lemma, Ergod. Th. Dynam. Sys., 30 (2010), 773-808.  doi: 10.1017/S0143385709000273.  Google Scholar

[10]

M. W. Hirsch, J. Palis, C. C. Pugh and M. Shub, Neighborhoods of hyperbolic sets, Inventiones Math., 9 (1969/70), 121-134.  doi: 10.1007/BF01404552.  Google Scholar

[11]

M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, Vol. 583. Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[12]

G. Liao, M. Viana and J. G. Yang, The entropy conjecture for diffeomorphisms away from tangencies, J. Eur. Math. Soc., 15 (2013), 2034-2060.  doi: 10.4171/JEMS/413.  Google Scholar

[13]

P. D. Liu and K. N. Lu, A note on partially hyperbolic attractors: Entropy conjecture and SRB measures, Discrete Contin. Dyn. Syst., 35 (2015), 341-352.  doi: 10.3934/dcds.2015.35.341.  Google Scholar

[14]

R. Mañé, An ergodic closing lemma, Ann. of Math., 116 (1982), 503-540.  doi: 10.2307/2007021.  Google Scholar

[15]

R. Mañé, Ergodic Theory and Differentiable Dynamics, Ergebnisse der Mathematik und Ihrer Grenzgebiete, 8. Springer-Verlag, Berlin, 1987. doi: 10.1007/978-3-642-70335-5.  Google Scholar

[16]

A. Manning, Topological entropy and the first homology group, Dynamical Systems - Warwick 1974, Lecture Notes in Math., Springer, Berlin, 468 (1975), 185-190.   Google Scholar

[17]

M. Misiurewicz, Diffeomorphisms without any measure of maximal entropy, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 21 (1973), 903-910.   Google Scholar

[18]

M. Misiurewicz, Topological conditional entropy, Studia Math., 55 (1976), 175-200.  doi: 10.4064/sm-55-2-175-200.  Google Scholar

[19]

K. Moriyasu, The ergodic closing lemma for C1 regular maps, Tokyo J. Math., 15 (1992), 172-183.  doi: 10.3836/tjm/1270130259.  Google Scholar

[20]

V. Pliss, A hypothesis due to Smale, Diff. Eq., 8 (1972), 203-214.   Google Scholar

[21]

C. C. Pugh, The closing lemma, Amer. J. Math., 89 (1967), 956-1009.  doi: 10.2307/2373413.  Google Scholar

[22]

C. C. Pugh and C. Robinson, The C1 closing lemma, including Hamiltonians, Ergod Th. Dynam. Sys., 3 (1983), 261-313.  doi: 10.1017/S0143385700001978.  Google Scholar

[23]

M. Qian and Z. S. Zhang, Ergodic theory for axiom A endomorphisms, Ergod. Th. Dynam. Sys., 15 (1995), 161-174.  doi: 10.1017/S0143385700008294.  Google Scholar

[24]

D. Ruelle and D. Sullivan, Currents, flows and diffeomorphisms, Topology, 14 (1975), 319-327.  doi: 10.1016/0040-9383(75)90016-6.  Google Scholar

[25]

R. Saghin and Z. H. Xia, The entropy conjecture for partially hyperbolic diffeomorphisms with 1-D center, Topology Appl., 157 (2010), 29-34.  doi: 10.1016/j.topol.2009.04.053.  Google Scholar

[26]

M. Shub, Dynamical systems, filtrations and entropy, Bull. Amer. Math. Soc., 80 (1974), 27-41.  doi: 10.1090/S0002-9904-1974-13344-6.  Google Scholar

[27]

M. Shub and R. F. Williams, Entropy and stability, Topology, 14 (1975), 329-338.  doi: 10.1016/0040-9383(75)90017-8.  Google Scholar

[28]

M. Urbanski and C. Wolf, SRB measures for Axiom A endomorphisms, Math. Res. Lett., 11 (2004), 785-797.  doi: 10.4310/MRL.2004.v11.n6.a6.  Google Scholar

[29]

P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79. Springer-Verlag, New York-Berlin, 1982.  Google Scholar

[30]

L. Wen, The C1 closing lemma for non-singular endomorphisms, Ergod. Th. Dynam. Sys., 11 (1991), 393-412.  doi: 10.1017/S0143385700006210.  Google Scholar

[31]

L. Wen, Generic diffeomorphisms away from homoclinic tangencies and heterodimensional cycles, Bull. Braz. Math. Soc. (N.S), 35 (2004), 419-452.  doi: 10.1007/s00574-004-0023-x.  Google Scholar

[32]

Y. Yomdin, Volume growth and entropy, Israel J. Math., 57 (1987), 285-300.  doi: 10.1007/BF02766215.  Google Scholar

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