Farey neighbors and hyperbolic Lorenz knots

TL;DR

This paper explores the conditions under which Lorenz knots, derived from symbolic dynamics of Lorenz maps, are hyperbolic, contingent on a conjecture and properties of Farey neighbors.

Contribution

It establishes a link between Farey neighbor sequences and the hyperbolicity of Lorenz knots, assuming a specific conjecture holds true.

Findings

Lorenz knots associated with certain Farey neighbor sequences are hyperbolic.

The result depends on the validity of a conjecture by Morton.

Provides a new criterion for hyperbolicity in Lorenz knots.

Abstract

Based on symbolic dynamics of Lorenz maps, we prove that, pro- vided one conjecture due to Morton is true, then Lorenz knots asso- ciated to orbits of points in the renormalization intervals of Lorenz maps with reducible kneading invariant of type (X,Y) * S, where the sequences X and Y are Farey neighbors verifying some conditions, are hyperbolic.