May  2006, 15(2): 611-640. doi: 10.3934/dcds.2006.15.611

Generic 3-dimensional volume-preserving diffeomorphisms with superexponential growth of number of periodic orbits

1. 

Mathematics 253-37, California Institute of Technology, Pasadena, CA, 91106, United States

2. 

Department of Mathematics, University of Toronto, Toronto, Ontario, M5S 2E4, Canada

Received  February 2005 Revised  December 2005 Published  March 2006

Let $M$ be a compact manifold of dimension three with a non-degenerate volume form $\Omega$ and Diff$^r_\Omega(M)$ be the space of $C^r$-smooth ($\Omega$-) volume-preserving diffeomorphisms of $M$ with $2\le r< \infty$. In this paper we prove two results. One of them provides the existence of a Newhouse domain $\mathcal N$ in Diff$^r_\Omega(M)$. The proof is based on the theory of normal forms [13], construction of certain renormalization limits, and results from [23], [26], [28], [32]. To formulate the second one, associate to each diffeomorphism a sequence $P_n(f)$ which gives for each $n$ the number of isolated periodic points of $f$ of period $n$. The main result of this paper states that for a Baire generic diffeomorphism $f$ in $\mathcal N$, the number of periodic points $P_n(f)$ grows with $n$ faster than any prescribed sequence of numbers $\{a_n\}_{n \in \mathbb Z_+}$ along a subsequence, i.e., $P_{n_i}(f)>a_{n_i}$ for some $n_i\to \infty$ with $i\to \infty$. The strategy of the proof is similar to the one of the corresponding $2$-dimensional non volume-preserving result [16]. The latter one is, in its turn, based on the Gonchenko-Shilnikov-Turaev Theorem [8], [9].
Citation: Vadim Kaloshin, Maria Saprykina. Generic 3-dimensional volume-preserving diffeomorphisms with superexponential growth of number of periodic orbits. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 611-640. doi: 10.3934/dcds.2006.15.611
[1]

Roland Zweimüller. Asymptotic orbit complexity of infinite measure preserving transformations. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 353-366. doi: 10.3934/dcds.2006.15.353

[2]

Rui Kuang, Xiangdong Ye. The return times set and mixing for measure preserving transformations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 817-827. doi: 10.3934/dcds.2007.18.817

[3]

S. Eigen, A. B. Hajian, V. S. Prasad. Universal skyscraper templates for infinite measure preserving transformations. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 343-360. doi: 10.3934/dcds.2006.16.343

[4]

Wenxiang Sun, Yun Yang. Hyperbolic periodic points for chain hyperbolic homoclinic classes. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3911-3925. doi: 10.3934/dcds.2016.36.3911

[5]

Tatiane C. Batista, Juliano S. Gonschorowski, Fábio A. Tal. Density of the set of endomorphisms with a maximizing measure supported on a periodic orbit. Discrete & Continuous Dynamical Systems - A, 2015, 35 (8) : 3315-3326. doi: 10.3934/dcds.2015.35.3315

[6]

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

[7]

Zhihong Xia. Homoclinic points and intersections of Lagrangian submanifold. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 243-253. doi: 10.3934/dcds.2000.6.243

[8]

Benjamin Wincure, Alejandro D. Rey. Growth regimes in phase ordering transformations. Discrete & Continuous Dynamical Systems - B, 2007, 8 (3) : 623-648. doi: 10.3934/dcdsb.2007.8.623

[9]

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.

[10]

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.

[11]

Oksana Koltsova, Lev Lerman. Hamiltonian dynamics near nontransverse homoclinic orbit to saddle-focus equilibrium. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 883-913. doi: 10.3934/dcds.2009.25.883

[12]

Benoît Grébert, Tiphaine Jézéquel, Laurent Thomann. Dynamics of Klein-Gordon on a compact surface near a homoclinic orbit. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3485-3510. doi: 10.3934/dcds.2014.34.3485

[13]

Shigui Ruan, Junjie Wei, Jianhong Wu. Bifurcation from a homoclinic orbit in partial functional differential equations. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1293-1322. doi: 10.3934/dcds.2003.9.1293

[14]

W.-J. Beyn, Y.-K Zou. Discretizations of dynamical systems with a saddle-node homoclinic orbit. Discrete & Continuous Dynamical Systems - A, 1996, 2 (3) : 351-365. doi: 10.3934/dcds.1996.2.351

[15]

K. H. Kim, F. W. Roush and J. B. Wagoner. Inert actions on periodic points. Electronic Research Announcements, 1997, 3: 55-62.

[16]

Charles Pugh, Michael Shub. Periodic points on the $2$-sphere. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1171-1182. doi: 10.3934/dcds.2014.34.1171

[17]

Peter Giesl. Converse theorem on a global contraction metric for a periodic orbit. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5339-5363. doi: 10.3934/dcds.2019218

[18]

Roland Gunesch, Anatole Katok. Construction of weakly mixing diffeomorphisms preserving measurable Riemannian metric and smooth measure. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 61-88. doi: 10.3934/dcds.2000.6.61

[19]

Nasab Yassine. Quantitative recurrence of some dynamical systems preserving an infinite measure in dimension one. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 343-361. doi: 10.3934/dcds.2018017

[20]

Sonja Hohloch. Transport, flux and growth of homoclinic Floer homology. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3587-3620. doi: 10.3934/dcds.2012.32.3587

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]