Periods implying almost all periods, trees with snowflakes, and zero entropy maps

الأصل · EN

Let X be a compact tree, f be a continuous map from X to itself, End(X) be the number of endpoints and Edg(X) be the number of edges of X. We show that if n>1 has no prime divisors less than End(X)+1 and f has a cycle of period n, then f has cycles of all periods greater than 2End(X)(n-1) and topological entropy h(f)>0; so if p is the least prime number greater than End(X) and f has cycles of all periods from 1 to 2End(X)(p-1), then f has cycles of all periods (this verifies a conjecture of Misiurewicz for tree maps). Together with the spectral decomposition theorem for graph maps it implies that h(f)>0 iff there exists n such that f has a cycle of period mn for any m. We also define snowflakes for tree maps and show that h(f)=0 iff every cycle of f is a snowflake or iff the period of every cycle of f is of form 2ˡm where m≤ Edg(X) is an odd integer with prime divisors less than End(X)+1.