American Institute of Mathematical Sciences

Advanced
July  2006, 14(3): 419-446. doi: 10.3934/dcds.2006.14.419

Unstable manifolds and Hölder structures associated with noninvertible maps

Eugen Mihailescu 1,

1. 

Institute of Mathematics "Simion Stoilow" of the Romanian Academy, P.O Box 1-764, RO 014-700, Bucharest, Romania

Received  January 2005 Revised  June 2005 Published  December 2005

We study the case of a smooth noninvertible map $f$ with Axiom A, in higher dimension. In this paper, we look first at the unstable dimension (i.e the Hausdorff dimension of the intersection between local unstable manifolds and a basic set $\Lambda$), and prove that it is given by the zero of the pressure function of the unstable potential, considered on the natural extension $\hat\Lambda$ of the basic set $\Lambda$; as a consequence, the unstable dimension is independent of the prehistory $\hat x$. Then we take a closer look at the theorem of construction for the local unstable manifolds of a perturbation $g$ of $f$, and for the conjugacy $\Phi_g$ defined on $\hat \Lambda$. If the map $g$ is holomorphic, one can prove some special estimates of the Hölder exponent of $\Phi_g$ on the liftings of the local unstable manifolds. In this way we obtain a new estimate of the speed of convergence of the unstable dimension of $g$, when $g \rightarrow f$. Afterwards we prove the real analyticity of the unstable dimension when the map $f$ depends on a real analytic parameter. In the end we show that there exist Gibbs measures on the intersections between local unstable manifolds and basic sets, and that they are in fact geometric measures; using this, the unstable dimension turns out to be equal to the upper box dimension. We notice also that in the noninvertible case, the Hausdorff dimension of basic sets does not vary continuously with respect to the perturbation $g$ of $f$. In the case of noninvertible Axiom A maps on $\mathbb P^2$, there can exist an infinite number of local unstable manifolds passing through the same point $x$ of the basic set $\Lambda$, thus there is no unstable lamination. Therefore many of the methods used in the case of diffeomorphisms break down and new phenomena and methods of proof must appear. The results in this paper answer to some questions of Urbanski ([21]) about the extension of one dimensional theory of Hausdorff dimension of fractals to the higher dimensional case. They also improve some results and estimates from [7].
Keywords: Gibbs states., unstable manifolds, topological pressure, Smale spaces, Hausdorff dimension.
Mathematics Subject Classification: Primary: 37D20, 37A35; Secondary: 37F3.
Citation: Eugen Mihailescu. Unstable manifolds and Hölder structures associated with noninvertible maps. Discrete & Continuous Dynamical Systems, 2006, 14 (3) : 419-446. doi: 10.3934/dcds.2006.14.419
[1]

Cleon S. Barroso. The approximate fixed point property in Hausdorff topological vector spaces and applications. Discrete & Continuous Dynamical Systems, 2009, 25 (2) : 467-479. doi: 10.3934/dcds.2009.25.467
[2]

Lin Shi, Dingshi Li, Kening Lu. Limiting behavior of unstable manifolds for spdes in varying phase spaces. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021020
[3]

Dongkui Ma, Min Wu. Topological pressure and topological entropy of a semigroup of maps. Discrete & Continuous Dynamical Systems, 2011, 31 (2) : 545-556. doi: 10.3934/dcds.2011.31.545
[4]

Eugen Mihailescu. Approximations for Gibbs states of arbitrary Hölder potentials on hyperbolic folded sets. Discrete & Continuous Dynamical Systems, 2012, 32 (3) : 961-975. doi: 10.3934/dcds.2012.32.961
[5]

Alexanger Arbieto, Carlos Arnoldo Morales Rojas. Topological stability from Gromov-Hausdorff viewpoint. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 3531-3544. doi: 10.3934/dcds.2017151
[6]

Michihiro Hirayama, Naoya Sumi. Hyperbolic measures with transverse intersections of stable and unstable manifolds. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1451-1476. doi: 10.3934/dcds.2013.33.1451
[7]

Eugen Mihailescu, Mariusz Urbański. Holomorphic maps for which the unstable manifolds depend on prehistories. Discrete & Continuous Dynamical Systems, 2003, 9 (2) : 443-450. doi: 10.3934/dcds.2003.9.443
[8]

C. M. Groothedde, J. D. Mireles James. Parameterization method for unstable manifolds of delay differential equations. Journal of Computational Dynamics, 2017, 4 (1&2) : 21-70. doi: 10.3934/jcd.2017002
[9]

Guizhen Cui, Yunping Jiang, Anthony Quas. Scaling functions and Gibbs measures and Teichmüller spaces of circle endomorphisms. Discrete & Continuous Dynamical Systems, 1999, 5 (3) : 535-552. doi: 10.3934/dcds.1999.5.535
[10]

De-Jun Feng, Antti Käenmäki. Equilibrium states of the pressure function for products of matrices. Discrete & Continuous Dynamical Systems, 2011, 30 (3) : 699-708. doi: 10.3934/dcds.2011.30.699
[11]

Hiroki Sumi, Mariusz Urbański. Bowen parameter and Hausdorff dimension for expanding rational semigroups. Discrete & Continuous Dynamical Systems, 2012, 32 (7) : 2591-2606. doi: 10.3934/dcds.2012.32.2591
[12]

Sara Munday. On Hausdorff dimension and cusp excursions for Fuchsian groups. Discrete & Continuous Dynamical Systems, 2012, 32 (7) : 2503-2520. doi: 10.3934/dcds.2012.32.2503
[13]

Shmuel Friedland, Gunter Ochs. Hausdorff dimension, strong hyperbolicity and complex dynamics. Discrete & Continuous Dynamical Systems, 1998, 4 (3) : 405-430. doi: 10.3934/dcds.1998.4.405
[14]

Luis Barreira and Jorg Schmeling. Invariant sets with zero measure and full Hausdorff dimension. Electronic Research Announcements, 1997, 3: 114-118.

[15]

Jon Chaika. Hausdorff dimension for ergodic measures of interval exchange transformations. Journal of Modern Dynamics, 2008, 2 (3) : 457-464. doi: 10.3934/jmd.2008.2.457
[16]

Krzysztof Barański, Michał Wardal. On the Hausdorff dimension of the Sierpiński Julia sets. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3293-3313. doi: 10.3934/dcds.2015.35.3293
[17]

Xinsheng Wang, Weisheng Wu, Yujun Zhu. Local unstable entropy and local unstable pressure for random partially hyperbolic dynamical systems. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 81-105. doi: 10.3934/dcds.2020004
[18]

Marc Rauch. Variational principles for the topological pressure of measurable potentials. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 367-394. doi: 10.3934/dcdss.2017018
[19]

Xueting Tian. Topological pressure for the completely irregular set of birkhoff averages. Discrete & Continuous Dynamical Systems, 2017, 37 (5) : 2745-2763. doi: 10.3934/dcds.2017118
[20]

M. Bulíček, Josef Málek, Dalibor Pražák. On the dimension of the attractor for a class of fluids with pressure dependent viscosities. Communications on Pure & Applied Analysis, 2005, 4 (4) : 805-822. doi: 10.3934/cpaa.2005.4.805

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (46)
  • HTML views (0)
  • Cited by (17)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences