zbMATH — the first resource for mathematics

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

28D05 Measure-preserving transformations
28A78 Hausdorff and packing measures
Full Text: DOI