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.,

[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.

show all references

References:
[1]

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

[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.

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
[1]

F. Rodriguez Hertz, M. A. Rodriguez Hertz, A. Tahzibi and R. Ures. A criterion for ergodicity for non-uniformly hyperbolic diffeomorphisms. Electronic Research Announcements, 2007, 14: 74-81. doi: 10.3934/era.2007.14.74

[2]

Jose F. Alves; Stefano Luzzatto and Vilton Pinheiro. Markov structures for non-uniformly expanding maps on compact manifolds in arbitrary dimension. Electronic Research Announcements, 2003, 9: 26-31.

[3]

Mark Pollicott. Local Hölder regularity of densities and Livsic theorems for non-uniformly hyperbolic diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 2005, 13 (5) : 1247-1256. doi: 10.3934/dcds.2005.13.1247

[4]

Nicolai T. A. Haydn, Kasia Wasilewska. Limiting distribution and error terms for the number of visits to balls in non-uniformly hyperbolic dynamical systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2585-2611. doi: 10.3934/dcds.2016.36.2585

[5]

Marzie Zaj, Abbas Fakhari, Fatemeh Helen Ghane, Azam Ehsani. Physical measures for certain class of non-uniformly hyperbolic endomorphisms on the solid torus. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1777-1807. doi: 10.3934/dcds.2018073

[6]

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

[7]

José F. Alves. Non-uniformly expanding dynamics: Stability from a probabilistic viewpoint. Discrete & Continuous Dynamical Systems - A, 2001, 7 (2) : 363-375. doi: 10.3934/dcds.2001.7.363

[8]

Rua Murray. Ulam's method for some non-uniformly expanding maps. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 1007-1018. doi: 10.3934/dcds.2010.26.1007

[9]

José F. Alves. A survey of recent results on some statistical features of non-uniformly expanding maps. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 1-20. doi: 10.3934/dcds.2006.15.1

[10]

Hartmut Schwetlick, Daniel C. Sutton, Johannes Zimmer. On the $\Gamma$-limit for a non-uniformly bounded sequence of two-phase metric functionals. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 411-426. doi: 10.3934/dcds.2015.35.411

[11]

Xueting Tian, Paulo Varandas. Topological entropy of level sets of empirical measures for non-uniformly expanding maps. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5407-5431. doi: 10.3934/dcds.2017235

[12]

Paul Wright. Differentiability of Hausdorff dimension of the non-wandering set in a planar open billiard. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3993-4014. doi: 10.3934/dcds.2016.36.3993

[13]

Fernando J. Sánchez-Salas. Dimension of Markov towers for non uniformly expanding one-dimensional systems. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1447-1464. doi: 10.3934/dcds.2003.9.1447

[14]

Thorsten Riess. Numerical study of secondary heteroclinic bifurcations near non-reversible homoclinic snaking. Conference Publications, 2011, 2011 (Special) : 1244-1253. doi: 10.3934/proc.2011.2011.1244

[15]

Yongluo Cao, Stefano Luzzatto, Isabel Rios. Some non-hyperbolic systems with strictly non-zero Lyapunov exponents for all invariant measures: Horseshoes with internal tangencies. Discrete & Continuous Dynamical Systems - A, 2006, 15 (1) : 61-71. doi: 10.3934/dcds.2006.15.61

[16]

Yan Huang. On Hausdorff dimension of the set of non-ergodic directions of two-genus double cover of tori. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2395-2409. doi: 10.3934/dcds.2018099

[17]

Alexander Arbieto, Luciano Prudente. Uniqueness of equilibrium states for some partially hyperbolic horseshoes. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 27-40. doi: 10.3934/dcds.2012.32.27

[18]

Fengjie Geng, Junfang Zhao, Deming Zhu, Weipeng Zhang. Bifurcations of a nongeneric heteroclinic loop with nonhyperbolic equilibria. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 133-145. doi: 10.3934/dcdsb.2013.18.133

[19]

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

[20]

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

2016 Impact Factor: 1.099

Metrics

  • PDF downloads (31)
  • HTML views (36)
  • Cited by (1)

Other articles
by authors

[Back to Top]