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Dimensions associated with recurrent self-similar sets. (English) Zbl 0742.28002

The authors prove that the probability measure associated with some recurrent self-similar set is exact-dimensional in the sense of C. D. Cutler [Ergodic Theory Dyn. Syst. 10, No. 3, 451-462 (1990; Zbl 0691.58023)] who has proved that an ergodic invariant measure \(\mu\) generated by some Lipschitz map on a metric space is exact-dimensional. This means that the dimension map \(x\to\limsup_{r\to 0}{\log\mu(B(x,r))\over \log r}\) is almost surely constant, where \(B(x,r)\) denotes the closed ball with centre at the point \(x\) and radius \(r\). The setting of the authors can be arranged to an ergodic dynamical system, and thus at least the first part of the result easily follows. Nevertheless, the concrete calculation of this dimension constant seems to be new. This generalizes the formula obtained by J. S. Geronimo and D. P. Hardin [Constructive Approximation 5, No.1, 89-98 (1989; Zbl 0666.28004)].
Furthermore, a recurrent iterated function system with affine maps in \(\mathbb{R}^ 2\) is considered for which the box dimension of its attractor \(A\) can be obtained from the dimension of the projection on the \(y\)-axis and some spectral radius condition resulting from the connection matrix (which describes the components of \(A\)) and the coefficients of the affine maps.

MSC:

28A80 Fractals
37A99 Ergodic theory
28A78 Hausdorff and packing measures
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