Attractors from one dimensional Lorenz-like maps
The author studies a collection of functions with following properties: (i) \(f: [a,b]\to [a,b]\); (ii) \(f\) has a jump discontinuity at a fixed-point \(c\in (a,b)\); (iii) \(f\) is \(C^1\) on \([a,b]\setminus\{c\}\) and there is a number \(\lambda> 1\) such that \(|f'(x)|> \lambda\) whenever \(f'(x)\) exists. Then there exists a compact invariant set \(L_f\subset [a,b]\), \(c\in L_f\), such that the restriction \(f|L_f\) is topologically transitive and \(W^s(L_f)= \{x\in [a,b]: f^n(x)\to L_f\}\) is open and dense. The author investigates the structure of the set \(L_f\) and gives an explicit construction of a trapping region \(N\) for \(L_f\). Finally, the properties, which are persistent with respect to continuous perturbations of \(f\), are investigated.
scientific article; zbMATH DE number 2110784