## Institute for Mathematical Sciences

## Preprint ims91-13

** A. M. Blokh**
* Periods Implying Almost All Periods, Trees with Snowflakes, and Zero Entropy Maps.*

Format: AmSTeX (version 1) Abstract: 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 $End(X)+1$.

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