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

Palis, 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 -