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

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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

Received: January 2017

Published: November 2017

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

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Abstract

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

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Mathematics Subject Classification: Primary: 37C29, 37E30; Secondary: 28D20.

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Figure 1. Local dynamics near a heteroclinic tangency

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Figure 2. Local dynamics near the parabolic tongues

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Figure 3. Simple composition of affine-like maps

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Figure 4. Parabolic composition of affine-like maps

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