"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). |
« back