LI-YORKE CHAOS FOR DENDRITE MAPS WITH ZERO TOPOLOGICAL ENTROPY AND ω-LIMIT SETS

Let X be a dendrite with set of endpoints E(X) closed and let f : X -> X be a continuous map with zero topological entropy. Let P( f) be the set of periodic points off and let L be an omega-limit set of f. We prove that if L is infinite then L boolean AND P(f) subset of E(X)', where E(X)' is the set of all accumulations points of E(X). Furthermore, if E(X) is countable and L is uncountable then L boolean AND P(f) = phi. We also show that if E(X)' is finite and L is uncountable then there is a sequence of subdendrites (D-k)k >= 1 of X and a sequence of integers n(k) >= 2 satisfying the following properties. For all k >= 1, 1. f(alpha k) (Dk) = Dk where alpha(k) = n(1)n(2) . . . n(k), 2. U-k(n)j (-1)(=0) f(k alpha j) 1 (D-j) subset of Dj-1 for all j >= 2, 3. L subset of boolean OR(alpha k-1) f(i)(Dk), 4. f(L boolean AND f(i)(Dk)) L boolean AND) f(i+1)(Dk) for any 0 <= i <= ak 1. In particular, L boolean AND fi(Dk) 0, 5. f (Dk) boolean AND f (Dk) has empty interior for any 0 <= i j <= alpha k, . As a consequence, if f has a Li-Yorke pair (x, y) with wf(x) or wf (y) uncountable then f is Li-Yorke chaotic.