Topological Entropy Of Interval Maps

In the present paper we study mainly the relations between topological entropies and chainable points sets of chain recurrent points, between topological entropies and property of characteristic 0, and between topological entropies and sequence equicon-tinuity of a continuous interval self-map.In Chapter One,we introduce briefly the historic background and notions of topological dynamical system and some known results about topological entropy.In Chapter Two, we discuss properties of the set of chainable points of a continuous self-map f on an impact metric space, especially those of chain recurrent points. We conclude that, for any chain recurrent point x of f, S(x, f), the set of chainable points of x under f, is a strongly invariant closed set; if S(x, f) is finite then it is equal to CE(x, f), the set of chain equivalent points of x.In Chapter Three, we study mainly equivalent conditions of a continuous interval map with positive topological entropies. Let f:I-I be a continuous self-map.We show that the following six statements are equivalent:(1)f has positive topological entropy;(2)there exists a non-periodic chain recurrent point x, such that S(x, f) has at least two minimal sets;(3) there exist a non-periodic chain recurrent point x and a positive integer t, such that S(x, ft) is not divisible under ft;(4)there exist a non-periodic chain recurrent point x and a positive integer t, such that S(x, ft) is equal to S(x, f2t);(5)for some sequence of positive integers S = {n1 < n2 < ...}, there exists a periodic point p, such that f|cR(f) is not S-equicontinuous at p;(6)there exists a periodic point p, such that f|cR(f) has not characteristic 0.