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

Raith, Peter

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].

Reviewer: Manfred Denker (Göttingen)

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

Keywords:

piecewise monotonic map of the interval; Misiurewicz condition; capacity; Hausdorff dimension; ergodic invariant probability measure

Citations:

Zbl 0770.28009; Zbl 0687.58013