The Hausdorff dimension of the set of dissipative points for a Cantor-like model set for singly cusped parabolic dynamics

TL;DR

This paper investigates a Cantor-like set within the unit interval and proves that both this set and its dissipative points have Hausdorff dimension 1, using properties of a non-symmetric Cauchy-type random walk.

Contribution

It introduces a specific Cantor-like set and establishes that its Hausdorff dimension and that of its dissipative points are both equal to 1, linking fractal geometry with stochastic processes.

Findings

Both the set and dissipative points have Hausdorff dimension 1

The proof employs the transience of a non-symmetric Cauchy-type random walk

The set is constructed within the context of parabolic dynamics

Abstract

In this paper we introduce and study a certain intricate Cantor-like set $C$ contained in unit interval. Our main result is to show that the set $C$ itself, as well as the set of dissipative points within $C$, both have Hausdorff dimension equal to 1. The proof uses the transience of a certain non-symmetric Cauchy-type random walk.