**
ACTA MATHEMATICA UNIVERSITATIS COMENIANAE **

Vol. 63, 1 (1994)

pp. 39-53

CONTINUITY OF THE HAUSDORFF DIMENSION FOR INVARIANT SUBSETS OF INTERVAL MAPS

P. RAITH

**Abstract**.
Let $T\:[0,1]\to [0,1]$ be an expanding piecewise monotonic map, and consider the set $R$ of all points, whose orbits omit a certain finite union of open intervals. It is shown that the Hausdorff dimension $\text HD\,(R)$ depends continuously on small perturbations of the endpoints of these open intervals. A similar result for the topological pressure is also obtained. Furthermore it is shown that for every $t\in [0,1]$ there exists a closed, $T$-invariant $R_t\subseteq [0,1]$ with $\text HD\,(R_t)=t$. Finally it is proved that the Hausdorff dimension of the set of all points, whose orbit is not dense, is $1$.

**AMS subject classification**.

**Keywords**.

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