February  2018, 38(2): 431-448. doi: 10.3934/dcds.2018020

An estimate on the Hausdorff dimension of stable sets of non-uniformly hyperbolic horseshoes

1. 

Université Paris 13, Sorbonne Paris Cité, LAGA, CNRS (UMR 7539), Villetaneuse, F-93439, France

2. 

IMPA, Instituto de Matemática Pura e Aplicada, 110, Estrada D. Castorina, CEP 22460-320, Rio de Janeiro, RJ, Brazil

* Corresponding author:Carlos Matheus

Received  January 2017 Published  February 2018

Fund Project: The authors were partially supported by the Balzan Research Project of J. Palis and the French ANR grand "DynPDE" (ANR-10-BLAN 0102).

We show that the Hausdorff dimension of stable sets of non-uniformly hyperbolic horseshoes is strictly smaller than two.

Citation: Carlos Matheus, Jacob Palis. An estimate on the Hausdorff dimension of stable sets of non-uniformly hyperbolic horseshoes. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 431-448. doi: 10.3934/dcds.2018020
References:
[1]

C. Matheus, J. Palis and J. -C. Yoccoz, The Hausdorff dimension of stable sets of non-uniformly hyperbolic horseshoes, work in progress. Google Scholar

[2]

J. Palis and J.-C. Yoccoz, Non-uniformly hyperbolic horseshoes arising from bifurcations of Poincaré heteroclinic cycles, Publ. Math. Inst. Hautes Études Sci., 110 (2009), 1-217.  doi: 10.1007/s10240-009-0023-x.  Google Scholar

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References:
[1]

C. Matheus, J. Palis and J. -C. Yoccoz, The Hausdorff dimension of stable sets of non-uniformly hyperbolic horseshoes, work in progress. Google Scholar

[2]

J. Palis and J.-C. Yoccoz, Non-uniformly hyperbolic horseshoes arising from bifurcations of Poincaré heteroclinic cycles, Publ. Math. Inst. Hautes Études Sci., 110 (2009), 1-217.  doi: 10.1007/s10240-009-0023-x.  Google Scholar

Figure 1.  Local dynamics near a heteroclinic tangency
Figure 2.  Local dynamics near the parabolic tongues
Figure 3.  Simple composition of affine-like maps
Figure 4.  Parabolic composition of affine-like maps
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