Chaotic behaviour of one-dimensional saddle-node horseshoes (Q1812307)

The author discusses the problem of chaotic dynamics persistence for a class of families \((\Psi_\mu)_\mu\) of one-dimensional maps exhibiting not only a critical point but also a saddle-node (neutral) fixed point, corresponding to the parameter \(\mu=0\). The main result: for the indicated class there exists a positive Lebesgue measure set \(S\) of parameter values such that for every \(\mu\in S\) the critical orbit has positive Lyapunov exponent; for almost \(\mu\in S\) the critical orbit is dense and so \(\Psi_\mu\) is transitive; moreover \(S\) is prevalent at \(\mu=0\).