# Dimension of Gibbs measures with infinite entropy

## TL;DR

This paper investigates the Hausdorff dimension of Gibbs measures with infinite entropy on interval maps with countably many branches, revealing their dimensional properties and providing explicit dimension values.

## Contribution

It establishes conditions under which such Gibbs measures are symbolic-exact dimensional and determines their local and Hausdorff dimensions.

## Key findings

- Gibbs measures with infinite entropy are symbolic-exact dimensional under certain conditions.
- The lower local dimension of these measures is almost surely zero.
- For the Gauss map, these measures have Hausdorff dimension zero and packing dimension 1/2.

## Abstract

We study the Hausdorff dimension of Gibbs measures with infinite entropy with respect to maps of the interval with countably many branches. We show that under simple conditions, such measures are symbolic-exact dimensional, and provide an almost sure value for the symbolic dimension. We also show that the lower local dimension dimension is almost surely equal to zero, while the upper local dimension is almost surely equal to the symbolic dimension. In particular, we prove that a large class of Gibbs measures with infinite entropy for the Gauss map have Hausdorff dimension zero and packing dimension equal to $1/2$, and so such measures are not exact dimensional.