How roundoff errors help to compute the rotation set of torus homeomorphisms

TL;DR

This paper investigates how finite precision and roundoff errors in computer calculations influence the approximation of the rotation set for torus homeomorphisms, proposing that coarse discretization yields better numerical results.

Contribution

It introduces the concepts of observable and asymptotic discretized rotation sets, demonstrating that coarse roundoff errors improve the numerical approximation of the rotation set.

Findings

Asymptotic discretized rotation set better approximates the true rotation set.

Coarse roundoff errors enhance numerical detection of the rotation set.

Theoretical and simulation results align on the importance of discretization.

Abstract

The goals of this paper are to obtain theoretical models of what happens when a computer calculates the rotation set of a homeomorphism, and to find a good algorithm to perform simulations of this rotation set. To do that we introduce the notion of observable rotation set, which takes into account the fact that we can only detect phenomenon appearing on positive Lebesgue measure sets; we also define the asymptotic discretized rotation set which in addition takes into account the fact that the computer calculates with a finite number of digits It appears that both theoretical results and simulations suggest that the asymptotic discretized rotation set is a much better approximation of the rotation set than the observable rotation set, in other words we need to do coarse roundoff errors to obtain numerically the rotation set.