A CHAOTIC FUNCTION WITH ZERO TOPOLOGICAL ENTROPY HAVING A NON-PERFECT ATTRACTOR
Bernd Kirchheim · Czech digital mathematics library · 1990
ABSRACT. In the paper there is constructed a continuous rnappingffrom a compact real interval to itself satisfying the following: The mapping f has zero topological entropy and there is an infinite attractor containing isolated points which are not approximate by periodic points. In the papers [4], [8] the existence of a continuous rnappingffrom the unit interval into itself with the following properties is remarked: a) The function fs of type 2X, that meansfhas no cycle of order not a power of 2 and has a cycle of any of the orders 1, 2, 22,... b) There is a point x such that vty(x) is infinite and has isolated points. (Here vty(x) denotes the set of all limit points of the trajectory {fn(x), n> 0} generated by x. Both examples in [4], [8] are not correct. In [3] a function having the described properties a, b) and satisfying moreover the condition that all isolated points of the infinite vty(x) belong to Per (f)\\Per (f) is constructed. (The existence of such an attractor is not explicitly remarked in [3] but follows from [6] where the