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doi: 10.3934/dcds.2021169
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On super-exponential divergence of periodic points for partially hyperbolic systems

Xiaolong Li 1,, and  Katsutoshi Shinohara 2,

1. 

Graduate School of Mathematics and Statistics, Huazhong University of Science and Technology, Luoyu Road 1037, Wuhan, China
2. 

Graduate School of Business Administration, Hitotsubashi University, 2-1 Naka, Kunitachi, Tokyo, Japan

* Corresponding author: Xiaolong Li

Received  January 2021 Revised  September 2021 Early access November 2021

We say that a diffeomorphism $ f $ is super-exponentially divergent if for every $ b>1 $ the lower limit of $ \#\mbox{Per}_n(f)/b^n $ diverges to infinity, where $ \mbox{Per}_n(f) $ is the set of all periodic points of $ f $ with period $ n $. This property is stronger than the usual super-exponential growth of the number of periodic points. We show that for any $ n $-dimensional smooth closed manifold $ M $ where $ n\ge 3 $, there exists a non-empty open subset $ \mathcal{O} $ of $ \mbox{Diff}^1(M) $ such that diffeomorphisms with super-exponentially divergent property form a dense subset of $ \mathcal{O} $ in the $ C^1 $-topology. A relevant result about the growth rate of the lower limit of the number of periodic points for diffeomorphisms in a $ C^r $-residual subset of $ \mbox{Diff}^r(M)\ (1\le r\le \infty) $ is also shown.

Keywords: Non-uniform hyperbolicity, partial hyperbolicity, heterodimensional cycles, super-exponential growth, number of periodic points.
Mathematics Subject Classification: Primary: 37C20, 37C25, 37C29, 37D30.
Citation: Xiaolong Li, Katsutoshi Shinohara. On super-exponential divergence of periodic points for partially hyperbolic systems. Discrete & Continuous Dynamical Systems, doi: 10.3934/dcds.2021169
References:
[1]

F. Abdenur, C. Bonatti, S. Crovisier, L. J. Díaz and L. Wen, Periodic points and homoclinic classes, Erg. Th. Dyn. Sys., 27 (2007), 1-22.  doi: 10.1017/S0143385706000538.  Google Scholar

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M. Asaoka, K. Shinohara and D. Turaev, Degenerate behavior in non-hyperbolic semi-group actions on the interval: Fast growth of periodic points and universal dynamics, Math. Ann., 368 (2017), 1277-1309.  doi: 10.1007/s00208-016-1468-0.  Google Scholar

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P. Berger, Generic family displaying robustly a fast growth of the number of periodic points, preprint, arXiv: 1701.02393. Google Scholar

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C. Bonatti and L. J. Díaz, Robust heterodimensional cycles and $C^1$-generic dynamics, J. Inst. Math. Jussieu, 7 (2008), 469-525.  doi: 10.1017/S1474748008000030.  Google Scholar

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C. Bonatti and L. J. Díaz, Fragile cycles, J. Differential Equations, 252 (2012), 4176-4199.  doi: 10.1016/j.jde.2011.12.002.  Google Scholar

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C. Bonatti, L. J. Díaz and T. Fisher, Super-exponential growth of the number of periodic orbits inside homoclinic classes, Discrete Contin. Dyn. Syst., 20 (2008), 589-604.  doi: 10.3934/dcds.2008.20.589.  Google Scholar

[9]

C. Bonatti, L. J. Díaz, E. R. Pujals and J. Rocha, Robustly transitive sets and heterodimensional cycles. Geometric methods in dynamics. I., Astérisque, 286 (2003), 187-222.   Google Scholar

[10]

C. Bonatti, L. J. Díaz and R. Ures, Minimality of strong stable and unstable foliations for partially hyperbolic diffeomorphisms, J. Inst. Math. Jussieu, 1 (2002), 513-541.  doi: 10.1017/S1474748002000142.  Google Scholar

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C. Bonatti, L. J. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity, Springer-Verlag, Berlin, 2005.  Google Scholar

[12] M. Brin and G. Stuck, Introduction to Dynamical Systems, Cambridge University Press, Cambridge, 2002.  doi: 10.1017/CBO9780511755316.  Google Scholar
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L. J. Díaz and R. Ures, Persistent homoclinic tangencies and the unfolding of cycles, Ann. Inst. H. Poincaré Anal. Non Linéaire, 11 (1994), 643-659.  doi: 10.1016/S0294-1449(16)30172-X.  Google Scholar

[14]

L. F. N. França, Partially hyperbolic sets with a dynamically minimal lamination, Discrete Contin. Dyn. Syst., 38 (2018), 2717-2729.  doi: 10.3934/dcds.2018114.  Google Scholar

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

[16]

S. Hayashi, Connecting invariant manifolds and the solution of the $C^1$ stability and $\Omega$-stability conjecture for flows, Ann. of Math., 145 (1997), 81-137.  doi: 10.2307/2951824.  Google Scholar

[17]

F. R. Hertz, M. R. Hertz and R. Ures, Some results on the integrability of the center bundle for partially hyperbolic diffeomorphisms, Partially Hyperbolic Dynamics, Laminations, and Teichmüller Flow, 51 (2007), 103-109.   Google Scholar

[18]

M. Hirsch, Differential Topology. Graduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg, 1976.  Google Scholar

[19]

V. Kaloshin, An extension of the Artin-Mazur theorem, Ann. Math., 150 (1999), 729-741.  doi: 10.2307/121093.  Google Scholar

[20]

K. Shinohara, An example of $C^1$-generically wild homoclinic classes with index deficiency, Nonlinearity, 24 (2011), 1961-1974.  doi: 10.1088/0951-7715/24/7/003.  Google Scholar

show all references

References:
[1]

F. Abdenur, C. Bonatti, S. Crovisier, L. J. Díaz and L. Wen, Periodic points and homoclinic classes, Erg. Th. Dyn. Sys., 27 (2007), 1-22.  doi: 10.1017/S0143385706000538.  Google Scholar

[2]

M. Artin and B. Mazur, On periodic points, Ann. of Math., 81 (1965), 82-99.  doi: 10.2307/1970384.  Google Scholar

[3]

M. Asaoka, K. Shinohara and D. Turaev, Degenerate behavior in non-hyperbolic semi-group actions on the interval: Fast growth of periodic points and universal dynamics, Math. Ann., 368 (2017), 1277-1309.  doi: 10.1007/s00208-016-1468-0.  Google Scholar

[4]

M. Asaoka, K. Shinohara and D. Turaev, Fast growth of the number of periodic points arising from heterodimensional connections, Compos. Math., 157 (2021), 1899-1963.  doi: 10.1112/S0010437X21007405.  Google Scholar

[5]

P. Berger, Generic family displaying robustly a fast growth of the number of periodic points, preprint, arXiv: 1701.02393. Google Scholar

[6]

C. Bonatti and L. J. Díaz, Robust heterodimensional cycles and $C^1$-generic dynamics, J. Inst. Math. Jussieu, 7 (2008), 469-525.  doi: 10.1017/S1474748008000030.  Google Scholar

[7]

C. Bonatti and L. J. Díaz, Fragile cycles, J. Differential Equations, 252 (2012), 4176-4199.  doi: 10.1016/j.jde.2011.12.002.  Google Scholar

[8]

C. Bonatti, L. J. Díaz and T. Fisher, Super-exponential growth of the number of periodic orbits inside homoclinic classes, Discrete Contin. Dyn. Syst., 20 (2008), 589-604.  doi: 10.3934/dcds.2008.20.589.  Google Scholar

[9]

C. Bonatti, L. J. Díaz, E. R. Pujals and J. Rocha, Robustly transitive sets and heterodimensional cycles. Geometric methods in dynamics. I., Astérisque, 286 (2003), 187-222.   Google Scholar

[10]

C. Bonatti, L. J. Díaz and R. Ures, Minimality of strong stable and unstable foliations for partially hyperbolic diffeomorphisms, J. Inst. Math. Jussieu, 1 (2002), 513-541.  doi: 10.1017/S1474748002000142.  Google Scholar

[11]

C. Bonatti, L. J. Díaz and M. Viana, Dynamics Beyond Uniform Hyperbolicity, Springer-Verlag, Berlin, 2005.  Google Scholar

[12] M. Brin and G. Stuck, Introduction to Dynamical Systems, Cambridge University Press, Cambridge, 2002.  doi: 10.1017/CBO9780511755316.  Google Scholar
[13]

L. J. Díaz and R. Ures, Persistent homoclinic tangencies and the unfolding of cycles, Ann. Inst. H. Poincaré Anal. Non Linéaire, 11 (1994), 643-659.  doi: 10.1016/S0294-1449(16)30172-X.  Google Scholar

[14]

L. F. N. França, Partially hyperbolic sets with a dynamically minimal lamination, Discrete Contin. Dyn. Syst., 38 (2018), 2717-2729.  doi: 10.3934/dcds.2018114.  Google Scholar

[15]

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

[16]

S. Hayashi, Connecting invariant manifolds and the solution of the $C^1$ stability and $\Omega$-stability conjecture for flows, Ann. of Math., 145 (1997), 81-137.  doi: 10.2307/2951824.  Google Scholar

[17]

F. R. Hertz, M. R. Hertz and R. Ures, Some results on the integrability of the center bundle for partially hyperbolic diffeomorphisms, Partially Hyperbolic Dynamics, Laminations, and Teichmüller Flow, 51 (2007), 103-109.   Google Scholar

[18]

M. Hirsch, Differential Topology. Graduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg, 1976.  Google Scholar

[19]

V. Kaloshin, An extension of the Artin-Mazur theorem, Ann. Math., 150 (1999), 729-741.  doi: 10.2307/121093.  Google Scholar

[20]

K. Shinohara, An example of $C^1$-generically wild homoclinic classes with index deficiency, Nonlinearity, 24 (2011), 1961-1974.  doi: 10.1088/0951-7715/24/7/003.  Google Scholar

Figure 1.  An illustration of an SH-simple cycle
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