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Bau-Sen Du introduced a notion of chaos which is stronger than LiYorke sensitivity. A TDS (X, f ) is called chaotic if there is a positive ε such that for any x and any nonempty open set V of X there is a point y in V such that the pair (x, y) is proximal but not ε-asymptotic. In this article, we show that a TDS (T, f ) is transitive but not mixing if and only if (T, f ) is Li-Yorke sensitive but not chaotic, where T is a tree. Moreover, we compare such chaos with other notions of chaos.

Topological entropy of nonautonomous piecewise monotone dynamical systems on the interval by Sergiȋ K o l y a d a (Kiev), Michał M i s i u r e w i c z (Indianapolis, Ind.) and L'ubomír S n o h a (Banská Bystrica) Dedicated to the memory of Wies law Szlenk

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