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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 |
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 and Continuous Dynamical Systems,
2022, 42
(4)
: 1707-1729.
doi: 10.3934/dcds.2021169
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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. |
| [2] |
M. Artin and B. Mazur,
On periodic points, Ann. of Math., 81 (1965), 82-99.
doi: 10.2307/1970384. |
| [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. |
| [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. |
| [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. 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. |
| [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.
|
| [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. |
| [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. 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.
|
| [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. 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. |
| [2] |
M. Artin and B. Mazur,
On periodic points, Ann. of Math., 81 (1965), 82-99.
doi: 10.2307/1970384. |
| [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. |
| [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. |
| [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. 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. |
| [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.
|
| [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. |
| [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. 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.
|
| [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. |
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