American Institute of Mathematical Sciences

Advanced
October  2009, 3(4): 555-587. doi: 10.3934/jmd.2009.3.555

Dynamics of the universal area-preserving map associated with period-doubling: Stable sets

Denis Gaidashev 1, and  Tomas Johnson 1,

1. 

Department of Mathematics, Uppsala University, Box 480, 751 06 Uppsala, Sweden, Sweden

Received  May 2009 Revised  November 2009 Published  January 2010

It is known that the famous Feigenbaum-Coullet-Tresser period-doubling universality has a counterpart for area-preserving maps of $R^2$. A renormalization approach was used in [11] and [12] in a computer-assisted proof of the existence of a "universal'' area-preserving map $F_*$, that is, a map with orbits of all binary periods $2^k, k \in N$. In this paper, we consider infinitely renormalizable maps, which are maps on the renormalization stable manifold in some neighborhood of $F_*$, and study their dynamics.
   For all such infinitely renormalizable maps in a neighborhood of the fixed point $F_*$, we prove the existence of a "stable'' invariant Cantor set $\l^\infty_F$ such that the Lyapunov exponents of $F |_{\l^\infty_F}$ are zero and whose Hausdorff dimension satisfies

$\text{dim}_H(\l_F^{\infty}) < 0.5324.$

   We also show that there exists a submanifold, $W_\omega$, of finite codimension in the renormalization local stable manifold such that for all $F\in W_\omega$, the set $\l^\infty_F$ is "weakly rigid'': the dynamics of any two maps in this submanifold, restricted to the stable set $\l^\infty_F$, are conjugate by a bi-Lipschitz transformation, which preserves the Hausdorff dimension.

Keywords: renormalization, rigidity, rigorous computations., Area-preserving maps, period-doubling.
Mathematics Subject Classification: 37E20, 37F25, 37D05, 37D20, 37C29, 37A05, 37G15, 37M9.
Citation: Denis Gaidashev, Tomas Johnson. Dynamics of the universal area-preserving map associated with period-doubling: Stable sets. Journal of Modern Dynamics, 2009, 3 (4) : 555-587. doi: 10.3934/jmd.2009.3.555
[1]

Denis Gaidashev, Tomas Johnson. Spectral properties of renormalization for area-preserving maps. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3651-3675. doi: 10.3934/dcds.2016.36.3651
[2]

Hans Koch. On hyperbolicity in the renormalization of near-critical area-preserving maps. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 7029-7056. doi: 10.3934/dcds.2016106
[3]

Simion Filip. Tropical dynamics of area-preserving maps. Journal of Modern Dynamics, 2019, 14: 179-226. doi: 10.3934/jmd.2019007
[4]

Mário Bessa, César M. Silva. Dense area-preserving homeomorphisms have zero Lyapunov exponents. Discrete and Continuous Dynamical Systems, 2012, 32 (4) : 1231-1244. doi: 10.3934/dcds.2012.32.1231
[5]

Giovanni Forni. The cohomological equation for area-preserving flows on compact surfaces. Electronic Research Announcements, 1995, 1: 114-123.

[6]

Daniel N. Dore, Andrew D. Hanlon. Area preserving maps on $\boldsymbol{S^2}$: A lower bound on the $\boldsymbol{C^0}$-norm using symplectic spectral invariants. Electronic Research Announcements, 2013, 20: 97-102. doi: 10.3934/era.2013.20.97
[7]

Àngel Jorba, Pau Rabassa, Joan Carles Tatjer. Period doubling and reducibility in the quasi-periodically forced logistic map. Discrete and Continuous Dynamical Systems - B, 2012, 17 (5) : 1507-1535. doi: 10.3934/dcdsb.2012.17.1507
[8]

H. Merdan, G. Caginalp. Decay of solutions to nonlinear parabolic equations: renormalization and rigorous results. Discrete and Continuous Dynamical Systems - B, 2003, 3 (4) : 565-588. doi: 10.3934/dcdsb.2003.3.565
[9]

Yiming Ding. Renormalization and $\alpha$-limit set for expanding Lorenz maps. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 979-999. doi: 10.3934/dcds.2011.29.979
[10]

Kaijen Cheng, Kenneth Palmer, Yuh-Jenn Wu. Period 3 and chaos for unimodal maps. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 1933-1949. doi: 10.3934/dcds.2014.34.1933
[11]

Miroslav KolÁŘ, Michal BeneŠ, Daniel ŠevČoviČ. Area preserving geodesic curvature driven flow of closed curves on a surface. Discrete and Continuous Dynamical Systems - B, 2017, 22 (10) : 3671-3689. doi: 10.3934/dcdsb.2017148
[12]

Jingzhi Yan. Existence of torsion-low maximal isotopies for area preserving surface homeomorphisms. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4571-4602. doi: 10.3934/dcds.2018200
[13]

Iuliana Oprea, Gerhard Dangelmayr. A period doubling route to spatiotemporal chaos in a system of Ginzburg-Landau equations for nematic electroconvection. Discrete and Continuous Dynamical Systems - B, 2019, 24 (1) : 273-296. doi: 10.3934/dcdsb.2018095
[14]

Tien-Cuong Dinh, Nessim Sibony. Rigidity of Julia sets for Hénon type maps. Journal of Modern Dynamics, 2014, 8 (3&4) : 499-548. doi: 10.3934/jmd.2014.8.499
[15]

H. E. Lomelí, J. D. Meiss. Generating forms for exact volume-preserving maps. Discrete and Continuous Dynamical Systems - S, 2009, 2 (2) : 361-377. doi: 10.3934/dcdss.2009.2.361
[16]

Marie Henry, Danielle Hilhorst, Masayasu Mimura. A reaction-diffusion approximation to an area preserving mean curvature flow coupled with a bulk equation. Discrete and Continuous Dynamical Systems - S, 2011, 4 (1) : 125-154. doi: 10.3934/dcdss.2011.4.125
[17]

Jianlu Zhang. Coexistence of period 2 and 3 caustics for deformative nearly circular billiard maps. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6419-6440. doi: 10.3934/dcds.2019278
[18]

Horst R. Thieme. Eigenvectors of homogeneous order-bounded order-preserving maps. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 1073-1097. doi: 10.3934/dcdsb.2017053
[19]

Rafael de la Llave, Jason D. Mireles James. Parameterization of invariant manifolds by reducibility for volume preserving and symplectic maps. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4321-4360. doi: 10.3934/dcds.2012.32.4321
[20]

Àngel Jorba, Pau Rabassa, Joan Carles Tatjer. Local study of a renormalization operator for 1D maps under quasiperiodic forcing. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 1171-1188. doi: 10.3934/dcdss.2016047

2020 Impact Factor: 0.848

Tools

Metrics

  • PDF downloads (67)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy