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

 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.

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:
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##### 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
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