7 research outputs found

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

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    Let XX be a compact tree, ff be a continuous map from XX to itself, End(X)End(X) be the number of endpoints and Edg(X)Edg(X) be the number of edges of XX. We show that if n>1n>1 has no prime divisors less than End(X)+1End(X)+1 and ff has a cycle of period nn, then ff has cycles of all periods greater than 2End(X)(n1)2End(X)(n-1) and topological entropy h(f)>0h(f)>0; so if pp is the least prime number greater than End(X)End(X) and ff has cycles of all periods from 1 to 2End(X)(p1)2End(X)(p-1), then ff 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)>0h(f)>0 iff there exists nn such that ff has a cycle of period mnmn for any mm. We also define {\it snowflakes} for tree maps and show that h(f)=0h(f)=0 iff every cycle of ff is a snowflake or iff the period of every cycle of ff is of form 2lm2^lm where mEdg(X)m\le Edg(X) is an odd integer with prime divisors less than End(X)+1End(X)+1
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