# Non-uniformly hyperbolic horseshoes arising from bifurcations of Poincaré heteroclinic cycles

Jacob Palis; Jean-Christophe Yoccoz

Publications Mathématiques de l'IHÉS (2009)

- Volume: 110, page 1-217
- ISSN: 0073-8301

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topPalis, Jacob, and Yoccoz, Jean-Christophe. "Non-uniformly hyperbolic horseshoes arising from bifurcations of Poincaré heteroclinic cycles." Publications Mathématiques de l'IHÉS 110 (2009): 1-217. <http://eudml.org/doc/273601>.

@article{Palis2009,

abstract = {In the present paper, we advance considerably the current knowledge on the topic of bifurcations of heteroclinic cycles for smooth, meaning C
∞, parametrized families \{g t
∣t∈ℝ\} of surface diffeomorphisms. We assume that a quadratic tangency q is formed at t=0 between the stable and unstable lines of two periodic points, not belonging to the same orbit, of a (uniformly hyperbolic) horseshoe K (see an example at the Introduction) and that such lines cross each other with positive relative speed as the parameter evolves, starting at t=0 and the point q. We also assume that, in some neighborhood W of K and of the orbit of tangency o(q), the maximal invariant set for g
0=g t=0 is K∪o(q), where o(q) denotes the orbit of q for g
0. We then prove that, when the Hausdorff dimension HD(K) is bigger than one, but not much bigger (see (H.4) in Section 1.2 for a precise statement), then for most t, |t| small, g t is a non-uniformly hyperbolic horseshoe in W, and so g t has no attractors in W. Most t, and thus most g t
, here means that t is taken in a set of parameter values with Lebesgue density one at t=0.},

author = {Palis, Jacob, Yoccoz, Jean-Christophe},

journal = {Publications Mathématiques de l'IHÉS},

keywords = {Poincaré heteroclinic cycles; bifurcation; hyperbolic horseshoe; Markov partition; Hausdorff dimension},

language = {eng},

pages = {1-217},

publisher = {Springer-Verlag},

title = {Non-uniformly hyperbolic horseshoes arising from bifurcations of Poincaré heteroclinic cycles},

url = {http://eudml.org/doc/273601},

volume = {110},

year = {2009},

}

TY - JOUR

AU - Palis, Jacob

AU - Yoccoz, Jean-Christophe

TI - Non-uniformly hyperbolic horseshoes arising from bifurcations of Poincaré heteroclinic cycles

JO - Publications Mathématiques de l'IHÉS

PY - 2009

PB - Springer-Verlag

VL - 110

SP - 1

EP - 217

AB - In the present paper, we advance considerably the current knowledge on the topic of bifurcations of heteroclinic cycles for smooth, meaning C
∞, parametrized families {g t
∣t∈ℝ} of surface diffeomorphisms. We assume that a quadratic tangency q is formed at t=0 between the stable and unstable lines of two periodic points, not belonging to the same orbit, of a (uniformly hyperbolic) horseshoe K (see an example at the Introduction) and that such lines cross each other with positive relative speed as the parameter evolves, starting at t=0 and the point q. We also assume that, in some neighborhood W of K and of the orbit of tangency o(q), the maximal invariant set for g
0=g t=0 is K∪o(q), where o(q) denotes the orbit of q for g
0. We then prove that, when the Hausdorff dimension HD(K) is bigger than one, but not much bigger (see (H.4) in Section 1.2 for a precise statement), then for most t, |t| small, g t is a non-uniformly hyperbolic horseshoe in W, and so g t has no attractors in W. Most t, and thus most g t
, here means that t is taken in a set of parameter values with Lebesgue density one at t=0.

LA - eng

KW - Poincaré heteroclinic cycles; bifurcation; hyperbolic horseshoe; Markov partition; Hausdorff dimension

UR - http://eudml.org/doc/273601

ER -

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