**"Stability of the topological pressure for continuous piecewise monotonic interval maps"**
#### Raith, PeterSuppose that $T:[0,1]\to [0,1]$ is
a piecewise monotonic map. This means there exists a
finite partition $\mathcal{Z}$ of $[0,1]$ into
pairwise disjoint open intervals whose union of their
closures equals $[0,1]$ such that $T\big|_Z$ is
continuous and strictly monotonic for
every $Z\in\mathcal{Z}$. Also continuously
differentiable piecewise monotonic maps will be
considered. For these classes of transformations two
topologies will be considered. One of them is the
$C^0$-topology but requiring that the maps have the
same number of intervals of monotonicity. The second
one is the $C^1$-topology allowing different numbers
of intervals of monotonicity but the number of
intervals of monotonicity bounded by a given
number $N$.
Continuity of the topological pressure will be
investigated. If $p(T,f)\geq\sup_{x\in [0,1]}f(x)$
then the topological pressure is lower
semi-continuous (in both topologies). Regarding the
$C^1$-topology one has always upper semi-continuity
of the topological pressure. For the $C^0$-topology
this is not the case in general. Upper bounds of the
“jumps up” are given. The topological pressure is
upper semi-continuous for all weight functions if and
only if there are no periodic points among the
endpoints of intervals of monotonicity (here $0$ and
$1$ are not considered as endpoints). |