Multifractal analysis of the convergence exponents for the digits in $d$-decaying Gauss like dynamical systems

TL;DR

This paper investigates the multifractal spectrum and Hausdorff dimensions related to digit convergence exponents in $d$-decaying Gauss-like dynamical systems, revealing complex fractal structures in digit sequences.

Contribution

It introduces a detailed multifractal analysis of convergence exponents and computes Hausdorff dimensions for intersections of digit-based sets in these systems.

Findings

Multifractal spectrum of convergence exponents characterized.

Hausdorff dimensions of specific digit sets calculated.

Intersections of digit sets exhibit intricate fractal properties.

Abstract

Let $\{a_n(x)\}_{n\geq1}$ be the sequence of digits of $x\in(0,1)$ in infinite iterated function systems with polynomial decay of the derivative. We first study the multifractal spectrum of the convergence exponent defined by the sequence of the digits $\{a_n(x)\}_{n\geq1}$ and the weighted products of distinct digits with finite numbers respectively, and then calculate the Hausdorff dimensions of the intersection of sets defined by the convergence exponent of the weighted product of distinct digits with finite numbers and sets of points whose digits are non-decreasing in such iterated function systems.