### Hausdorff dimension for piecewise monotonic maps

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If $f:[0,1]\to \mathbb{R}$ is strictly increasing and continuous define ${T}_{f}x=f\left(x\right)\phantom{\rule{0.166667em}{0ex}}(mod1)$. A transformation $\tilde{T}:[0,1]\to [0,1]$ is called $\epsilon $-close to ${T}_{f}$, if $\tilde{T}x=\tilde{f}\left(x\right)\phantom{\rule{0.166667em}{0ex}}(mod1)$ for a strictly increasing and continuous function $\tilde{f}:[0,1]\to \mathbb{R}$ with $\parallel \tilde{f}{-f\parallel}_{\infty}<\epsilon $. It is proved that the topological pressure $p({T}_{f},g)$ is lower semi-continuous, and an upper bound for the jumps up is given. Furthermore the continuity of the maximal measure is shown, if a certain condition is satisfied. Then it is proved that the topological pressure is upper semi-continuous for every continuous function $g:[0,1]\to \mathbb{R}$, if and only if $0$ is...

In this paper piecewise monotonic maps $T[0,1]\to [0,1]$ are considered. Let $Q$ be a finite union of open intervals, and consider the set $R\left(Q\right)$ of all points whose orbits omit $Q$. The influence of small perturbations of the endpoints of the intervals in $Q$ on the dynamical system $\left(R\right(Q),T)$ is investigated. The decomposition of the nonwandering set into maximal topologically transitive subsets behaves very unstably. Nonetheless, it is shown that a maximal topologically transitive subset cannot be completely destroyed by arbitrary...

Transformations T:[0,1] → [0,1] with two monotonic pieces are considered. Under the assumption that T is topologically transitive and ${h}_{top}\left(T\right)>0$, it is proved that the invariant measures concentrated on periodic orbits are dense in the set of all invariant probability measures.

The multifractal generalizations of Hausdorff dimension and packing dimension are investigated for an invariant subset A of a piecewise monotonic map on the interval. Formulae for the multifractal dimension of an ergodic invariant measure, the essential multifractal dimension of A, and the multifractal Hausdorff dimension of A are derived.

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