Deviations in the Franks-Misiurewicz conjecture

TL;DR

This paper investigates the Franks-Misiurewicz conjecture, demonstrating that any counterexample in the rational case would need to have unbounded deviations in a specific direction, thus providing insight into the conjecture's structure.

Contribution

It establishes a necessary condition for counterexamples in the rational case, linking unbounded deviations to the conjecture's potential failure.

Findings

Counterexamples must have unbounded deviations in the complementary direction of their rotation set.

Provides a new criterion to identify or rule out potential counterexamples.

Advances understanding of the structure of rotation sets in dynamical systems.

Abstract

We show that if there exists a counter example for the rational case of the Franks-Misiurewicz conjecture, then it must exhibit unbounded deviations in the complementary direction of its rotation set.