# Set-oriented numerical computation of rotation sets

## TL;DR

This paper introduces a set-oriented numerical algorithm for approximating the rotation set of certain homeomorphisms on the two-torus, with proven convergence and error estimates.

## Contribution

It develops a novel set-oriented method for computing rotation sets, including theoretical convergence proofs and error bounds, advancing numerical analysis in dynamical systems.

## Key findings

- Algorithm converges as precision and iteration increase
- Error estimates are provided under boundedness assumptions
- Method is applicable to systems with non-empty interior rotation sets

## Abstract

We establish a set-oriented algorithm for the numerical approximation of the rotation set of homeomorphisms of the two-torus homotopic to the identity. A theoretical background is given by the concept of {\epsilon}-rotation sets. These are obtained by replacing orbits with {\epsilon}-pseudo-orbits in the definition of the Misiurewicz-Ziemian rotation set and are shown to converge to the latter as {\epsilon} decreases to zero. Based on this result, we prove the convergence of the numerical approximations as precision and iteration time tend to infinity. Further, we provide analytic error estimates for the algorithm under an additional boundedness assumption, which is known to hold in many relevant cases and in particular for non-empty interior rotation sets.