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

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 {\it 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 $2^lm$ where $m\le Edg(X)$ is an odd integer with prime divisors less than $End(X)+1$

Dynamical consequences of a free interval: minimality, transitivity, mixing and topological entropy

We study dynamics of continuous maps on compact metrizable spaces containing a free interval (i.e., an open subset homeomorphic to an open interval). A special attention is paid to relationships between topological transitivity, weak and strong topological mixing, dense periodicity and topological entropy as well as to the topological structure of minimal sets. In particular, a trichotomy for minimal sets and a dichotomy for transitive maps are proved.Comment: 21 page