Measures and their dimension spectrums for cookie-cutter sets in \(\mathbb{R}^d\)
Cookie-cutter sets have been intensively studied either as fractal repellers of smooth expansive dynamical systems (cookie-cutters) or as invariant sets of related nonlinear contracting iterated function systems. In this paper, fractal geometric theory of cookie-cutter sets in \(n\)-dimensional Euclidean space is derived assuming certain conditions. These conditions basically ensure that the so-called bounded distortion property holds so that the methods developed in [\textit{T. Bedford}, in: Fractal geometry and analysis, NATO ASI Ser., Ser. C 346, 1-44 (1991; Zbl 0741.58011)] for the one-dimensional case apply in higher dimensions. Specifically, the paper extends to its higher dimensional setting: (i) Bowen's pressure formula for the Hausdorff dimension of the invariant set, (ii) existence and natural properties, including multifractal decomposition, of self-similar, conformal and Gibbs measures, and (iii) existence of constant order-two density. The situation considered in the paper includes self-similar sets and self-conformal sets as particular cases, for which most of the results in the paper are well-known [\textit{K. J. Falconer}, ``Techniques in fractal geometry'' (1998; Zbl 0869.28003)]. In [\textit{L. M. Barreira}, Ergodic Theory Dyn. Syst. 16, No. 5, 871-927 (1996; Zbl 0862.58042)], dimension theory for hyperbolic sets is developed in a broad setting that includes more general classes of non-conformal repellers.
scientific article; zbMATH DE number 1462489