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arXiv · 1007.1500

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

Shin Kiriki·Ming-Chia Li·Teruhiko Soma
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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 $J'$ such that, for any $a\in J'$, $f_{a,b}$ 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).

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Shin Kiriki, Ming-Chia Li, Teruhiko Soma. 2010-07-09. Coexistence of invariant sets with and without SRB measures in Hénon family. https://doi.org/10.1088/0951-7715%2F23%2F9%2F010

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