# Fixed points of the Berezin transform of polyanlytic Fock spaces

## TL;DR

This paper characterizes the fixed points of the Berezin transform in polyanalytic Fock spaces, showing they are harmonic functions and that the only bounded fixed points are constants.

## Contribution

It provides a complete description of fixed points of the Berezin transform in polyanalytic Fock spaces, linking them to harmonic functions and identifying constant functions as the only bounded fixed points.

## Key findings

- Fixed points are harmonic functions.
- Bounded fixed points are constant functions.
- Invariant functions under the Berezin transform are characterized.

## Abstract

We study the fixed points of the Berezin transform in polyanalytic Fock spaces of $\mathbb{C}$. We show that an $L^p$ function, $p\in[1,+\infty]$, with respect to the Lebesgue measure is invariant under this transformation if and only if it is harmonic. From this we deduce that the only bounded fixed points of the Berezin transform of polyanalytic Fock spaces are constant functions.