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Capacity for expanding piecewise monotonic interval maps. (English) Zbl 0941.28014

Förg-Rob, W. (ed.) et al., Iteration theory. Proceedings of the European conference, ECIT ’92, Batschuns, Austria, September 13-19, 1992. Singapore: World Scientific. 224-235 (1996).
Let \(T\) be a piecewise monotonic expanding map on a union \(X\) of finite intervals and set \(R(T)= \bigcap_{n\geq 0} T^{-n}X\). Assuming that \(T'\) is piecewise Hölder and that \(T\) satisfies the Misiurewicz condition, it is shown that the capacity and the Hausdorff dimension of \(R(T)\) coincide (Theorem 2). Moreover, the same result holds for the capacity and Hausdorff dimension of an ergodic invariant probability measure (Theorem 3). Related results appeared by F. Hofbauer and P. Raith [Can. Math. Bull. 35, No. 1, 84-98 (1992; Zbl 0770.28009)] and P. Raith [Stud. Math. 94, No. 1, 17-33 (1989; Zbl 0687.58013)].
For the entire collection see [Zbl 0901.00041].

MSC:

28D05 Measure-preserving transformations
37E05 Dynamical systems involving maps of the interval
28A78 Hausdorff and packing measures
26A18 Iteration of real functions in one variable
54H05 Descriptive set theory (topological aspects of Borel, analytic, projective, etc. sets)
37A05 Dynamical aspects of measure-preserving transformations
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