April  2022, 42(4): 1707-1729. doi: 10.3934/dcds.2021169

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 Published  April 2022 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 and Continuous Dynamical Systems, 2022, 42 (4) : 1707-1729. doi: 10.3934/dcds.2021169
References:
[1]

F. AbdenurC. BonattiS. CrovisierL. J. Díaz and L. Wen, Periodic points and homoclinic classes, Erg. Th. Dyn. Sys., 27 (2007), 1-22.  doi: 10.1017/S0143385706000538.

[2]

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

[3]

M. AsaokaK. 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.

[4]

M. AsaokaK. 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.

[5]

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

[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.

[7]

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

[8]

C. BonattiL. 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.

[9]

C. BonattiL. J. DíazE. R. Pujals and J. Rocha, Robustly transitive sets and heterodimensional cycles. Geometric methods in dynamics. I., Astérisque, 286 (2003), 187-222. 

[10]

C. BonattiL. 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.

[11]

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

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

[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.

[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.

[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.

[17]

F. R. HertzM. 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. 

[18]

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

[19]

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

[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.

show all references

References:
[1]

F. AbdenurC. BonattiS. CrovisierL. J. Díaz and L. Wen, Periodic points and homoclinic classes, Erg. Th. Dyn. Sys., 27 (2007), 1-22.  doi: 10.1017/S0143385706000538.

[2]

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

[3]

M. AsaokaK. 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.

[4]

M. AsaokaK. 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.

[5]

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

[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.

[7]

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

[8]

C. BonattiL. 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.

[9]

C. BonattiL. J. DíazE. R. Pujals and J. Rocha, Robustly transitive sets and heterodimensional cycles. Geometric methods in dynamics. I., Astérisque, 286 (2003), 187-222. 

[10]

C. BonattiL. 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.

[11]

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

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

[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.

[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.

[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.

[17]

F. R. HertzM. 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. 

[18]

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

[19]

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

[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.

Figure 1.  An illustration of an SH-simple cycle
[1]

Christian Bonatti, Lorenzo J. Díaz, Todd Fisher. Super-exponential growth of the number of periodic orbits inside homoclinic classes. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 589-604. doi: 10.3934/dcds.2008.20.589

[2]

Yakov Pesin, Vaughn Climenhaga. Open problems in the theory of non-uniform hyperbolicity. Discrete and Continuous Dynamical Systems, 2010, 27 (2) : 589-607. doi: 10.3934/dcds.2010.27.589

[3]

Boris Hasselblatt, Yakov Pesin, Jörg Schmeling. Pointwise hyperbolicity implies uniform hyperbolicity. Discrete and Continuous Dynamical Systems, 2014, 34 (7) : 2819-2827. doi: 10.3934/dcds.2014.34.2819

[4]

Mickaël Kourganoff. Uniform hyperbolicity in nonflat billiards. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1145-1160. doi: 10.3934/dcds.2018048

[5]

Sergey Kryzhevich, Sergey Tikhomirov. Partial hyperbolicity and central shadowing. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 2901-2909. doi: 10.3934/dcds.2013.33.2901

[6]

Luis Barreira, Claudia Valls. Growth rates and nonuniform hyperbolicity. Discrete and Continuous Dynamical Systems, 2008, 22 (3) : 509-528. doi: 10.3934/dcds.2008.22.509

[7]

Vadim Yu. Kaloshin and Brian R. Hunt. A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms II. Electronic Research Announcements, 2001, 7: 28-36.

[8]

Vadim Yu. Kaloshin and Brian R. Hunt. A stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms I. Electronic Research Announcements, 2001, 7: 17-27.

[9]

Andy Hammerlindl, Jana Rodriguez Hertz, Raúl Ures. Ergodicity and partial hyperbolicity on Seifert manifolds. Journal of Modern Dynamics, 2020, 0: 331-348. doi: 10.3934/jmd.2020012

[10]

Federico Rodriguez Hertz, María Alejandra Rodriguez Hertz, Raúl Ures. Partial hyperbolicity and ergodicity in dimension three. Journal of Modern Dynamics, 2008, 2 (2) : 187-208. doi: 10.3934/jmd.2008.2.187

[11]

Jérôme Buzzi, Todd Fisher. Entropic stability beyond partial hyperbolicity. Journal of Modern Dynamics, 2013, 7 (4) : 527-552. doi: 10.3934/jmd.2013.7.527

[12]

Yakov Pesin. On the work of Dolgopyat on partial and nonuniform hyperbolicity. Journal of Modern Dynamics, 2010, 4 (2) : 227-241. doi: 10.3934/jmd.2010.4.227

[13]

David Damanik, Jake Fillman, Milivoje Lukic, William Yessen. Characterizations of uniform hyperbolicity and spectra of CMV matrices. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 1009-1023. doi: 10.3934/dcdss.2016039

[14]

Andy Hammerlindl. Partial hyperbolicity on 3-dimensional nilmanifolds. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3641-3669. doi: 10.3934/dcds.2013.33.3641

[15]

Eleonora Catsigeras, Xueting Tian. Dominated splitting, partial hyperbolicity and positive entropy. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 4739-4759. doi: 10.3934/dcds.2016006

[16]

Rafael Potrie. Partial hyperbolicity and foliations in $\mathbb{T}^3$. Journal of Modern Dynamics, 2015, 9: 81-121. doi: 10.3934/jmd.2015.9.81

[17]

Marcin Mazur, Jacek Tabor, Piotr Kościelniak. Semi-hyperbolicity and hyperbolicity. Discrete and Continuous Dynamical Systems, 2008, 20 (4) : 1029-1038. doi: 10.3934/dcds.2008.20.1029

[18]

Zhong-Jie Han, Gen-Qi Xu. Exponential decay in non-uniform porous-thermo-elasticity model of Lord-Shulman type. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 57-77. doi: 10.3934/dcdsb.2012.17.57

[19]

Dawei Yang, Shaobo Gan, Lan Wen. Minimal non-hyperbolicity and index-completeness. Discrete and Continuous Dynamical Systems, 2009, 25 (4) : 1349-1366. doi: 10.3934/dcds.2009.25.1349

[20]

Boris Kalinin, Victoria Sadovskaya. Normal forms for non-uniform contractions. Journal of Modern Dynamics, 2017, 11: 341-368. doi: 10.3934/jmd.2017014

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (188)
  • HTML views (177)
  • Cited by (0)

Other articles
by authors

[Back to Top]