Coexistence of invariant sets with and without SRB measures in H\'enon family

TL;DR

This paper investigates the coexistence of invariant sets with and without SRB measures in the Hénon family, revealing complex dynamical behaviors including homoclinic tangencies, Newhouse phenomena, and simultaneous existence of different types of attractors.

Contribution

It demonstrates the existence of parameter intervals where both SRB and non-SRB invariant sets coexist in the Hénon family, extending understanding of complex dynamical phenomena.

Findings

Existence of dense parameter subsets with homoclinic tangencies.

Presence of residual sets exhibiting Newhouse phenomena.

Simultaneous existence of large non-SRB sets and small SRB attractors.

Abstract

Let $\{f_{a,b}\}$ be the (original) H\'enon 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).