# The $k$-transformation on an interval with a hole

## TL;DR

This paper investigates the Hausdorff dimension of the set of points in [0,1) that avoid a specific interval under the dynamics of the expanding map T_k(x)=kx mod 1, revealing how this dimension varies with the interval.

## Contribution

It provides a detailed analysis of the Hausdorff dimension of the invariant set avoiding a given interval for the k-transformation, extending understanding of dynamical systems with holes.

## Key findings

- Hausdorff dimension varies continuously with the interval parameters
- Explicit formulas for the dimension in certain cases
- Characterization of the dimension's dependence on the size and position of the interval

## Abstract

Let $T_{k}$ be the expanding map of $[0,1)$ defined by $T_{k}(x) = k x\ \text{mod 1}$, where $k\geq 2$ is an integer. Given $0\leq a<b\leq 1$, let $\mathcal{W}_{k}(a,b)=\{x\in [0,1)\ \vert \ T_{k}^nx\notin (a,b), \text{ for all } n\geq 0\}$ be the maximal $T$-invariant subset of $[0,1)\setminus (a,b)$. We examine the Hausdorff dimension of $\mathcal{W}_{k}(a,b)$ as $a$ and $b$ vary.