# Invariant curves of smooth quasi-periodic mappings

## TL;DR

This paper investigates the existence of invariant curves in smooth quasi-periodic planar mappings, focusing on mappings with multiple frequencies and specific smoothness conditions.

## Contribution

It establishes conditions for the existence of invariant curves in quasi-periodic mappings with high smoothness and multiple frequencies, extending previous results in dynamical systems.

## Key findings

- Proves existence of invariant curves under certain smoothness and intersection conditions.
- Extends invariant curve theory to mappings with multiple frequencies.
- Provides mathematical criteria for invariant curve existence in quasi-periodic systems.

## Abstract

In this paper we are concerned with the existence of invariant curves of planar mappings which are quasi-periodic in the spatial variable, satisfy the intersection property, $\mathcal{C}^{p}$ smooth with $p>2n+1$, $n$ is the number of frequencies.