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Chaotic functions with zero topological entropy. (English) Zbl 0639.54029
In “Period three implies chaos”, Am. Math. Monthly 82, 985-992 (1975; Zbl 0351.92021) T. Y. Li and J. A. Yorke introduced a notion of chaos for mappings of a compact interval to itself. Mappings with positive topological entropy exhibit his type of chaotic behaviour. The author is concerned with maps with zero topological entropy. S is called a $$\delta$$-scrambled set for f for some $$\delta\geq 0$$ if for any x,y$$\in S$$ $$x\neq y$$, and any periodic point p we have $\limsup_{n\to \infty}| f^ n(x)-f^ n(y)| >\delta,\quad \liminf_{n\to \infty}| f^ n(x)-f^ n(y)| =0\quad and\quad \limsup_{n\to \infty}| f^ n(x)-f^ n(p)| >\delta.$ f is chaotic if it has an infinite scrambled set. Any map with a cycle whose order is not a power of 2 is chaotic, while maps which do not have such a cycle either have cycles of finitely many different orders which are powers of 2 or have cycles of infinitely many such orders. The latter classes are said to be of type $$2^{\infty}$$. The properties of such maps are studied. The main result is a characterization of when they are chaotic and the relationship with infinite attractors (the limit points of trajectories). Finally an example is given of a $$2^{\infty}$$ map which is chaotic and an example is given of a $$2^{\infty}$$ map which has an infinite attractor but which is non chaotic.
Reviewer: M.Sears

##### MSC:
 54H20 Topological dynamics (MSC2010) 26A18 Iteration of real functions in one variable 37C75 Stability theory for smooth dynamical systems 37A99 Ergodic theory
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