# Recognition of symmetries in reversible maps

## TL;DR

This paper investigates the symmetries of reversible germs of diffeomorphisms, revealing that their reversing symmetry groups are generally infinite and providing geometric insights into their discrete dynamics, especially in the generic case.

## Contribution

It establishes that reversing symmetry groups of reversible germs are typically infinite, contrasting with continuous dynamics, and introduces geometric tools to analyze their fixed-point structures.

## Key findings

- Reversing symmetry groups are generally infinite.
- Fixed-point subspaces provide geometric insights into dynamics.
- Results apply to generic cases with transversal linear involutions.

## Abstract

We deal with germs of diffeomorphisms that are reversible under an involution. We establish that this condition implies that, in general, both the family of reversing symmetries and the group of symmetries are not finite, in contrast with continuous-time dynamics, where typically there are finitely many reversing symmetries. From this we obtain two chains of fixed-points subspaces of involutory reversing symmetries that we use to obtain geometric information on the discrete dynamics generated by a given diffeomorphism. The results are illustrated by the generic case in arbitrary dimension, when the diffeomorphism is the composition of transversal linear involutions.