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On a dimension of measures. (English) Zbl 1020.28004

The authors introduce a notion of dimension – the concentration dimension – for a Borel probability measure \(\mu\) on a metric space \(X\), as follows \[ \underline{\dim}_L\mu=\underline{\lim}_{r\rightarrow 0}{\log Q_\mu(r)\over \log r}, \overline{\dim}_L\mu=\overline{\lim}_{r\rightarrow 0}{\log Q_\mu(r)\over\log r}, \] where \(Q_{\mu}(r)=\sup\{\mu B(x,r):x\in X\}\) is the so-called Lévy concentration function. They show that concentration dimensions are comparable to the correlation dimensions (it holds that \(1\leq {\underline{\dim}_C\mu\over \underline{\dim}_L\mu}\leq 2\) and also for upper versions) and that lower concentration dimension bounds Hausdorff dimension from below (\(\dim_L\mu\leq \dim_H\mu\)). They also compute lower and upper bounds for the concentration dimensions of invariant measures for iterated function systems with place-dependent probabilities and apply these results to invariant measures (1) for a class of semifractals [A. Lasota and J. Myjak, “Fractals, semifractals and Markov operators”, Int. J. Bifurcation Chaos Appl. Sci. Eng. 9, No. 2, 301-325 (1999; Zbl 0973.28004)] and (2) for the semiflow of a first-order partial differential equation governing blood cell growth [A. Lasota, M. C. Mackey and M. Ważewska-Czyzewska, “Minimizing therapeutically induced anemia”, J. Math. Biol. 13, 149-158 (1981; Zbl 0473.92003)].

MSC:

28A80 Fractals
37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
28A78 Hausdorff and packing measures
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