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Local dimension for piecewise monotonic maps on the interval. (English) Zbl 0842.58019
The author considers piecewise monotonic maps of the interval and examines the pointwise dimension of measures. For a measure \(m\), the pointwise dimension \(l_m(x)\) at a point \(x\) is given by the expression \(\lim_{r\downarrow 0} \log m(B_r(x))/\log r\) where this exists. This quantity is closely related to the Hausdorff dimension of a measure and in particular, if \(l_m(x)\) is constant almost everywhere with respect to \(m\) then the Hausdorff dimension of \(m\) is this constant value. The author shows that for piecewise monotonic maps which have derivatives of bounded \(p\)-variation, the pointwise dimension of an ergodic invariant measure with positive Lyapunov exponent \(\chi_m = \int \log |T'|dm\) is given by \(h_m/\chi_m\) almost everywhere, where \(h_m\) is the entropy of the map with respect to \(m\). He goes on to discuss the multifractal properties of conformal measures of expanding maps. Defining \(L_m(a)\) to be \(\{x : l_m(x) = a\}\), he considers the Hausdorff dimension of the sets \(L_m(a)\). He shows that under some basic conditions, the quantity \(HD(L_m(a))\) is a concave function of \(a\) for \(a\) in a closed bounded interval, this function arising as the Legendre transform of a pressure functional. In the course of the proof, the author extends the notion of topological pressure to deal with a larger class of functions than the traditional class of continuous functions.
Reviewer: A.Quas (Cambridge)

37E99 Low-dimensional dynamical systems
37A99 Ergodic theory
54H20 Topological dynamics (MSC2010)
28A78 Hausdorff and packing measures
Full Text: DOI
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