A class of measures and non-stationary fractals, associated to f-expansions

TL;DR

This paper constructs and analyzes non-stationary Moran fractals related to f-expansions, studying their Hausdorff dimensions and dependence on number representations using ergodic theory and pressure functions.

Contribution

It introduces a new class of non-stationary Moran fractals linked to f-expansions and derives their Hausdorff dimensions as zeros of pressure functions, with analytical dependence on frequencies.

Findings

Hausdorff dimension equals zero of a pressure function

Dimensions depend analytically on asymptotic frequencies

Fractals associated with various f-expansions studied in detail

Abstract

We construct first a class of Moran fractals in R^d with countably many generators and non-stationary contraction rates; at each step n, the contractions depend on n-truncated sequences, and are related to asymptotic letter frequencies. In some cases the sets of contractions may be infinite at each step. We show that the Hausdorff dimension of such a fractal is equal to the zero h of a pressure function. We prove that the dimensions of these sets depend real analytically on the frequencies. Next, we apply the above construction to obtain non-stationary fractals E(x; f) \subset R^d, associated to f-expansions of real numbers x, and study the dependence of these fractals on x. We consider for instance beta-expansions, the continued fraction expansion and other f-expansions. By employing the Ergodic Theorem for invariant absolutely continuous measures and equilibrium measures, and using some probabilities for which the digits become independent random variables, we study the function x \to dim_H(E(x; f)) on the respective set of quasinormal numbers x \in [0; 1). We investigate also another class of fractals \tilde E_f (x) \subset R^d, for which both the non-stationary contraction vectors and the asymptotic frequencies depend on the f-representation of x. We obtain then some properties of the digits of x, related to \tilde E_f (x) and to equilibrium measures.