Strange attractors in saddle-node cycles: Prevalence and globality
The paper deals with parametrized families of diffeomorphisms bifurcating through the creation of critical saddle-node cycles. The authors prove that: 1. If \(\{f_\mu\}_\mu\) is a generic smooth family of diffeomorphisms on a manifold \(M\), unfolding a critical saddle-node cycle and \(A\) is the set of all values of \(\mu\) for which \(f_\mu\) exhibits Hénon-like strange attractors then \[ \liminf_{\varepsilon\to 0} \frac{m(A\cap (-\varepsilon,\varepsilon))}{2\varepsilon}>0. \] 2. Let \(B\) be the set of values of \(\mu\) for which \[ \overline{\bigcup_{x\in M}\alpha(x)}\cup \overline{\bigcup_{x\in M} \omega(x)} \] is a hyperbolic set of \(f_\mu\). Then, there exists an open set of smooth families of diffeomorphisms \(\{f_\mu\}_{\mu\in\mathbb{R}}\) unfolding a critical saddle-node cycle, for which \(B\) satisfies \[ \liminf_{\varepsilon\to\infty} \frac{m(B\cap (-\varepsilon, \varepsilon))}{2\varepsilon}>0. \]
scientific article; zbMATH DE number 885520