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The Hausdorff dimension of an ergodic invariant measure for a piecewise monotonic map of the interval. (English) Zbl 0701.28005
A map $$T$$ on the interval $$[0,1]$$ is called piecewise monotonic, if there exists a finite or countable set $${\mathcal Z}$$ of pairwise disjoint open subintervals of $$[0,1]$$ such that $$T| Z$$ is strictly monotone and continuous for all $$Z\in {\mathcal Z}$$. Such a map $$T$$ is said to have a derivative of bounded p-variation, if there exists a function $$g:\;[0,1]\to {\mathbb{R}}$$ with $$var^ p_{[0,1]}g<\infty$$ and $$g(x)=0$$ for $$x\in [0,1]\setminus \cup_{Z\in {\mathcal Z}}Z$$ such that $$T| Z$$ is an antiderivative $$g| Z$$ for $$Z\in {\mathcal Z}$$. The Hausdorff dimension $$HD(\mu)$$ of a probability measure $$\mu$$ is definedas the infimum of the Hausdorff dimensions $$HD(X)$$, the infimum taken over all $$X\subset [0,1]$$ with $$\mu(X)=1$$. If now $$T$$ is a piecewise monotonic map on [0,1] with finite $${\mathcal Z}$$ and a derivative $$T'$$ of bounded $$p$$-variation for some $$p>0$$, it is shown for an ergodic invariant measure $$\mu$$ concentrated on $$\cup_{Z\in {\mathcal Z}}Z$$, that $$HD(\mu)=h_{\mu}/\lambda_{\mu},$$ provided that the Lyapunov exponent $$\lambda_{\mu}=\int \log | T'| d\mu$$ is strictly positive. The proofs show that this result also holds, if instead of assuming that $${\mathcal Z}$$ is finite, one assumes that the set of all $$x$$, which satisfy $$x\in T(Z)$$ for infinitely many $$Z\in {\mathcal Z}$$, is at most countable.
Reviewer: F.Hofbauer

##### MSC:
 28D05 Measure-preserving transformations 28A78 Hausdorff and packing measures
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