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Spectral decomposition for rescaling expansive flows with rescaled shadowing
A forward Ergodic Closing Lemma and the Entropy Conjecture for nonsingular endomorphisms away from tangencies
Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Tokyo, Japan |
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.
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. |
[2] |
R. Bowen,
Entropy expansive maps, Trans. Amer. Math. Soc., 164 (1972), 323-331.
doi: 10.1090/S0002-9947-1972-0285689-X. |
[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. |
[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. |
[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. |
[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. |
[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. |
[9] |
S. Hayashi,
An extension of the ergodic closing lemma, Ergod. Th. Dynam. Sys., 30 (2010), 773-808.
doi: 10.1017/S0143385709000273. |
[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. |
[11] |
M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, Vol. 583. Springer-Verlag, Berlin-New York, 1977. |
[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. |
[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. |
[14] |
R. Mañé,
An ergodic closing lemma, Ann. of Math., 116 (1982), 503-540.
doi: 10.2307/2007021. |
[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. |
[16] |
A. Manning,
Topological entropy and the first homology group, Dynamical Systems - Warwick 1974, Lecture Notes in Math., Springer, Berlin, 468 (1975), 185-190.
|
[17] |
M. Misiurewicz,
Diffeomorphisms without any measure of maximal entropy, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 21 (1973), 903-910.
|
[18] |
M. Misiurewicz,
Topological conditional entropy, Studia Math., 55 (1976), 175-200.
doi: 10.4064/sm-55-2-175-200. |
[19] |
K. Moriyasu,
The ergodic closing lemma for C1 regular maps, Tokyo J. Math., 15 (1992), 172-183.
doi: 10.3836/tjm/1270130259. |
[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. |
[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. |
[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. |
[24] |
D. Ruelle and D. Sullivan,
Currents, flows and diffeomorphisms, Topology, 14 (1975), 319-327.
doi: 10.1016/0040-9383(75)90016-6. |
[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. |
[26] |
M. Shub,
Dynamical systems, filtrations and entropy, Bull. Amer. Math. Soc., 80 (1974), 27-41.
doi: 10.1090/S0002-9904-1974-13344-6. |
[27] |
M. Shub and R. F. Williams,
Entropy and stability, Topology, 14 (1975), 329-338.
doi: 10.1016/0040-9383(75)90017-8. |
[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. |
[29] |
P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79. Springer-Verlag, New York-Berlin, 1982. |
[30] |
L. Wen,
The C1 closing lemma for non-singular endomorphisms, Ergod. Th. Dynam. Sys., 11 (1991), 393-412.
doi: 10.1017/S0143385700006210. |
[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. |
[32] |
Y. Yomdin,
Volume growth and entropy, Israel J. Math., 57 (1987), 285-300.
doi: 10.1007/BF02766215. |
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. |
[2] |
R. Bowen,
Entropy expansive maps, Trans. Amer. Math. Soc., 164 (1972), 323-331.
doi: 10.1090/S0002-9947-1972-0285689-X. |
[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. |
[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. |
[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. |
[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. |
[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. |
[9] |
S. Hayashi,
An extension of the ergodic closing lemma, Ergod. Th. Dynam. Sys., 30 (2010), 773-808.
doi: 10.1017/S0143385709000273. |
[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. |
[11] |
M. W. Hirsch, C. C. Pugh and M. Shub, Invariant Manifolds, Lecture Notes in Mathematics, Vol. 583. Springer-Verlag, Berlin-New York, 1977. |
[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. |
[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. |
[14] |
R. Mañé,
An ergodic closing lemma, Ann. of Math., 116 (1982), 503-540.
doi: 10.2307/2007021. |
[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. |
[16] |
A. Manning,
Topological entropy and the first homology group, Dynamical Systems - Warwick 1974, Lecture Notes in Math., Springer, Berlin, 468 (1975), 185-190.
|
[17] |
M. Misiurewicz,
Diffeomorphisms without any measure of maximal entropy, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 21 (1973), 903-910.
|
[18] |
M. Misiurewicz,
Topological conditional entropy, Studia Math., 55 (1976), 175-200.
doi: 10.4064/sm-55-2-175-200. |
[19] |
K. Moriyasu,
The ergodic closing lemma for C1 regular maps, Tokyo J. Math., 15 (1992), 172-183.
doi: 10.3836/tjm/1270130259. |
[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. |
[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. |
[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. |
[24] |
D. Ruelle and D. Sullivan,
Currents, flows and diffeomorphisms, Topology, 14 (1975), 319-327.
doi: 10.1016/0040-9383(75)90016-6. |
[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. |
[26] |
M. Shub,
Dynamical systems, filtrations and entropy, Bull. Amer. Math. Soc., 80 (1974), 27-41.
doi: 10.1090/S0002-9904-1974-13344-6. |
[27] |
M. Shub and R. F. Williams,
Entropy and stability, Topology, 14 (1975), 329-338.
doi: 10.1016/0040-9383(75)90017-8. |
[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. |
[29] |
P. Walters, An Introduction to Ergodic Theory, Graduate Texts in Mathematics, 79. Springer-Verlag, New York-Berlin, 1982. |
[30] |
L. Wen,
The C1 closing lemma for non-singular endomorphisms, Ergod. Th. Dynam. Sys., 11 (1991), 393-412.
doi: 10.1017/S0143385700006210. |
[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. |
[32] |
Y. Yomdin,
Volume growth and entropy, Israel J. Math., 57 (1987), 285-300.
doi: 10.1007/BF02766215. |
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