Coexistence of invariant sets with and without SRB measures in Hénon family

Abstract

Let $\{f_{a,b}\}$ be the (original) Hénon family. In this paper, we show that, for any $b$ near $0$, there exists a closed interval $J_b$ which contains a dense subset $J^{\prime }$ such that, for any $a\in J^{\prime }$, $f_{a,b}$ has a quadratic homoclinic tangency associated with a saddle fixed point of $f_{a,b}$ which unfolds generically with respect to the one-parameter family $\{f_{a,b}{\}}_{a\in J_b}$. By applying this result, we prove that $J_b$ contains a residual subset $A_b^{(2)}$ such that, for any $a\in A_b^{(2)}$, $f_{a,b}$ admits the Newhouse phenomenon. Moreover, the interval $J_b$ contains a dense subset ${\tilde{A}}_b$ such that, for any $a\in {\tilde{A}}_b$, $f_{a,b}$ has a large homoclinic set without SRB measure and a small strange attractor with SRB measure simultaneously. Dedicated to the memory of Floris Takens (Nov. 12, 1940 - Jun. 20, 2010).

arXiv commentTo appear in Nonlinearity (2010)

References in corpus (1)

• Persistent antimonotonic bifurcations and strange attractors for cubic homoclinic tangencies

Cited by in corpus (2)

• Takens' last problem and existence of non-trivial wandering domains
• Nonexistence of Wandering Domains for Infinitely Renormalizable Hénon Maps