On the dimension of deterministic and random Cantor-like sets, symbolic dynamics, and the Eckmann-Ruelle conjecture
This paper unifies and extends many of known results on the dimension of deterministic and random Cantor-like sets in \(\mathbb{R}^n\). The authors verify the Eckmann-Ruelle conjecture for equilibrium measures for Hölder continuous conformal expanding maps and conformal Axiom \(A^\#\) homeomorphism. They also show that the non-conformal Hölder continuous version of the conjecture is false. The Cantor-like sets will be constructed by arbitrary symbolic dynamical systems. The original idea of Moran will be extended to much broader classes of geometric constructions. The basic Moran-like construction starts with a compact shift invariant subset \(Q\) of an infinite product space of \(p\) symbols. The basic geometric sets \(\Delta_{i_1,\dots,i_n}\) in \(\mathbb{R}^n\), where \((i_1,\dots,i_n,i_{n+1}',\dots)\in Q\), have an inner and an outer closed ball, such that the inner balls for fixed \(n\) are disjoint, and the radii of both types equal to the product of the ratio coefficients associated to the symbols \(i_1,\dots,i_n\) (up to two positive constants).
scientific article; zbMATH DE number 1002420
scientific article