Work

Better estimates of Hölder thickness of fractals

Year: 2025

Type: article

Abstract: Dimensions of level sets of generic continuous functions and generic Hölder functions defined on a fractal F encode information about the geometry, “the thickness” of F . While in the continuous case this quantity is related to a reasonably tame dimension notion which is called the topological Hausdorff dimension of F , the Hölder case seems to be highly nontrivial. A number of earlier papers attempted to deal with this problem, carrying out investigation in the case of Hausdorff dimension and box dimension. In this paper we continue our study of the Hausdorff dimension of almost every level set of generic 1 -Hölder- \alpha functions, denoted by D_{*}(\alpha, F) . We substantially improve previous lower and upper bounds on D_{*}(\alpha, \Delta) , where \Delta is the Sierpiński triangle, achieving asymptotically equal bounds as \alpha\to 0+ . Using a similar argument, we also give an even stronger lower bound on the generic lower box dimension of level sets. Finally, we construct a connected fractal F on which there is a phase transition of D_{*}(\alpha, F) , thus providing the first example exhibiting this behaviour.

Source: Journal of Fractal Geometry Mathematics of Fractals and Related Topics

DOI: 10.4171/jfg/174

Authors: Zoltán Buczolich, Balázs Maga, Gáspár Vértesy

Institutions: Eötvös Loránd University, HUN-REN Alfréd Rényi Institute of Mathematics

Language: English

Topic: Mathematical Dynamics and Fractals

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: diamond

Funders: Nemzeti Kutatási Fejlesztési és Innovációs Hivatal, Innovációs és Technológiai Minisztérium, Nemzeti Kutatási, Fejlesztési és Innovaciós Alap, National Research, Development and Innovation Office