American Institute of Mathematical Sciences

Advanced
August  2008, 21(3): 907-928. doi: 10.3934/dcds.2008.21.907

Transversal families of hyperbolic skew-products

Eugen Mihailescu 1, and  Mariusz Urbański 2,

1. 

Institute of Mathematics "Simion Stoilow" of the Romanian Academy, P.O Box 1-764, RO 014-700, Bucharest
2. 

Department of Mathematics, University of North Texas, P.O. Box 311430, Denton, TX 76203-1430

Received  June 2007 Revised  January 2008 Published  April 2008

We study families of hyperbolic skew products with the transversality condition and in particular, the Hausdorff dimension of their fibers, by using thermodynamical formalism. The maps we consider can be non-invertible, and the study of their dynamics is influenced greatly by this fact.
    We introduce and employ probability measures (constructed from equilibrium measures on the natural extension), which are supported on the fibers of the skew product. A stronger condition, that of Uniform Transversality is then considered in order to obtain a general formula for Hausdorff dimension of fibers for all base points and almost all parameters.
    In the end we study a large class of examples of transversal hyperbolic families which locally depend linearly on the parameters, and also another class of examples related to complex dynamics.
Keywords: skew products, Hausdorff dimension, hyperbolic maps, transversality..
Mathematics Subject Classification: Primary: 37D35, 37D20; Secondary: 37A3.
Citation: Eugen Mihailescu, Mariusz Urbański. Transversal families of hyperbolic skew-products. Discrete and Continuous Dynamical Systems, 2008, 21 (3) : 907-928. doi: 10.3934/dcds.2008.21.907
[1]

C.P. Walkden. Stable ergodicity of skew products of one-dimensional hyperbolic flows. Discrete and Continuous Dynamical Systems, 1999, 5 (4) : 897-904. doi: 10.3934/dcds.1999.5.897
[2]

Davit Karagulyan. Hausdorff dimension of a class of three-interval exchange maps. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1257-1281. doi: 10.3934/dcds.2020077
[3]

Vanderlei Horita, Marcelo Viana. Hausdorff dimension for non-hyperbolic repellers II: DA diffeomorphisms. Discrete and Continuous Dynamical Systems, 2005, 13 (5) : 1125-1152. doi: 10.3934/dcds.2005.13.1125
[4]

Carlos Matheus, Jacob Palis. An estimate on the Hausdorff dimension of stable sets of non-uniformly hyperbolic horseshoes. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 431-448. doi: 10.3934/dcds.2018020
[5]

Anatole Katok. Hyperbolic measures and commuting maps in low dimension. Discrete and Continuous Dynamical Systems, 1996, 2 (3) : 397-411. doi: 10.3934/dcds.1996.2.397
[6]

David Färm, Tomas Persson. Dimension and measure of baker-like skew-products of $\boldsymbol{\beta}$-transformations. Discrete and Continuous Dynamical Systems, 2012, 32 (10) : 3525-3537. doi: 10.3934/dcds.2012.32.3525
[7]

Roy Adler, Bruce Kitchens, Michael Shub. Stably ergodic skew products. Discrete and Continuous Dynamical Systems, 1996, 2 (3) : 349-350. doi: 10.3934/dcds.1996.2.349
[8]

Rui Gao, Weixiao Shen. Analytic skew-products of quadratic polynomials over Misiurewicz-Thurston maps. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 2013-2036. doi: 10.3934/dcds.2014.34.2013
[9]

Roy Adler, Bruce Kitchens, Michael Shub. Errata to "Stably ergodic skew products". Discrete and Continuous Dynamical Systems, 1999, 5 (2) : 456-456. doi: 10.3934/dcds.1999.5.456
[10]

Matthieu Astorg, Fabrizio Bianchi. Higher bifurcations for polynomial skew products. Journal of Modern Dynamics, 2022, 18: 69-99. doi: 10.3934/jmd.2022003
[11]

Àlex Haro. On strange attractors in a class of pinched skew products. Discrete and Continuous Dynamical Systems, 2012, 32 (2) : 605-617. doi: 10.3934/dcds.2012.32.605
[12]

Jose S. Cánovas, Antonio Falcó. The set of periods for a class of skew-products. Discrete and Continuous Dynamical Systems, 2000, 6 (4) : 893-900. doi: 10.3934/dcds.2000.6.893
[13]

Matúš Dirbák. Minimal skew products with hypertransitive or mixing properties. Discrete and Continuous Dynamical Systems, 2012, 32 (5) : 1657-1674. doi: 10.3934/dcds.2012.32.1657
[14]

Viorel Nitica. Examples of topologically transitive skew-products. Discrete and Continuous Dynamical Systems, 2000, 6 (2) : 351-360. doi: 10.3934/dcds.2000.6.351
[15]

Wen Huang, Jianya Liu, Ke Wang. Möbius disjointness for skew products on a circle and a nilmanifold. Discrete and Continuous Dynamical Systems, 2021, 41 (8) : 3531-3553. doi: 10.3934/dcds.2021006
[16]

Jon Aaronson, Michael Bromberg, Nishant Chandgotia. Rational ergodicity of step function skew products. Journal of Modern Dynamics, 2018, 13: 1-42. doi: 10.3934/jmd.2018012
[17]

Hiroki Sumi, Mariusz Urbański. Bowen parameter and Hausdorff dimension for expanding rational semigroups. Discrete and Continuous Dynamical Systems, 2012, 32 (7) : 2591-2606. doi: 10.3934/dcds.2012.32.2591
[18]

Sara Munday. On Hausdorff dimension and cusp excursions for Fuchsian groups. Discrete and Continuous Dynamical Systems, 2012, 32 (7) : 2503-2520. doi: 10.3934/dcds.2012.32.2503
[19]

Shmuel Friedland, Gunter Ochs. Hausdorff dimension, strong hyperbolicity and complex dynamics. Discrete and Continuous Dynamical Systems, 1998, 4 (3) : 405-430. doi: 10.3934/dcds.1998.4.405
[20]

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

2021 Impact Factor: 1.588

Tools

Metrics

  • PDF downloads (95)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy