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DCDS-B

It is well known that a continuous piecewise monotone
interval map with positive topological entropy is semiconjugate to a
map of a constant slope and the same entropy, and if it is
additionally transitive then this semiconjugacy is actually a
conjugacy. We generalize this result to
piecewise continuous piecewise monotone interval maps, and
as a consequence, get it also for piecewise monotone graph maps.
We show that assigning to a continuous transitive piecewise
monotone map of positive entropy a map of constant slope conjugate to
it defines an operator, and show that this operator is not continuous.

DCDS

In this paper we give a partial characterization of the periodic
tree patterns of maximum entropy for a given period. More precisely,
we prove that each periodic pattern with maximal entropy is
irreducible (has no block structures) and simplicial (any vertex
belongs to the periodic orbit). Moreover, we also prove that it is
maximodal in the sense that every point of the periodic orbit is a
"turning point".

DCDS

We study the set of periods of degree 1 continuous maps from $\sigma$
into itself, where $\sigma$ denotes the space shaped like the letter $\sigma$
(i.e., a segment attached to a circle by one of its endpoints).
Since the maps under consideration have degree 1, the rotation theory can
be used.
We show that, when the interior of the rotation interval contains an
integer, then the set of periods (of periodic points of any rotation
number) is the set of all integers except maybe $1$ or $2$.
We exhibit degree 1 $\sigma$-maps $f$ whose set of periods is a combination
of the set of periods of a degree 1 circle map and the set of periods of
a $3$-star (that is, a space shaped like the letter $Y$).
Moreover, we study the set of periods forced by periodic orbits that do
not intersect the circuit of $\sigma$; in particular, when there exists
such a periodic orbit whose diameter (in the covering space) is at least $1$,
then there exist periodic points of all periods.

DCDS

We prove that, given a tree pattern $\mathcal{P}$, the set of
periods of a minimal representative $f: T\rightarrow T$ of $\mathcal{P}$ is contained
in the set of periods of any other representative. This statement is an
immediate corollary of the following stronger result: there is a
period-preserving injection from the set of periodic points of $f$ into that
of any other representative of $\mathcal{P}$. We prove this result by
extending the main theorem of [6] to negative cycles.

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