# Unique Ergodicity For Infinite Area Translation Surfaces

## TL;DR

This paper proves that for typical infinite staircase translation surfaces and certain wind-tree models, the translation flow is uniquely ergodic in almost every direction, contrasting with the non-ergodic behavior on periodic surfaces.

## Contribution

It establishes unique ergodicity for infinite area translation surfaces with typical configurations, extending understanding of ergodic properties in infinite translation surfaces.

## Key findings

- Unique ergodicity holds for typical infinite staircase surfaces.
- The result applies to the Ehrenfest wind-tree model with Hausdorff topology.
- Periodic translation surfaces cannot be uniquely ergodic in any direction.

## Abstract

We consider infinite staircase translation surfaces with varying step sizes. For typical step sizes we show that the translation flow is uniquely ergodic in almost every direction. Our result also hold for typical configurations of the Ehrenfest wind-tree model endowed with the Hausdorff topology. In contrast, we show that the translation flow on a periodic translation surface can not be uniquely ergodic in any direction.