# Spatial limit theorem for interval exchange transformations

## TL;DR

This paper establishes a spatial limit theorem for generic interval exchange transformations, showing that normalized ergodic sums behave asymptotically like ergodic integrals of translation flows on flat surfaces, revealing deep statistical properties.

## Contribution

It introduces a spatial limit theorem for IETs, linking their ergodic sums to translation flow integrals, a novel connection in dynamical systems.

## Key findings

- Normalized ergodic sums follow the same distributional asymptotics as translation flow integrals.
- The result applies to generic IETs and sufficiently regular functions.
- Provides a new statistical description of IET behavior.

## Abstract

We prove a spatial limit theorem for generic interval exchange transformations (IETs): for a generic IET the normalized ergodic sums of a sufficiently regular (e.g., Lipschitz) function have the same asymptotic behavior of distributions as the behaviour of ergodic integrals for a generic translation flow on a flat surface, described by A. Bufetov.