Invisible Tricorns in Real Slices of Rational Maps

TL;DR

This paper investigates the existence and properties of invisible Tricorn-type hyperbolic components in real slices of bicritical rational maps, providing topological characterizations and proving their abundance in certain parameter spaces.

Contribution

It introduces a topological criterion to identify invisible Tricorn components and proves their infinite presence in specific families of real bicritical maps.

Findings

Infinite invisible Tricorn components exist in studied families.

Topological properties characterize invisibility.

Methods apply broadly to generic real bicritical maps.

Abstract

One of the conspicuous features of real slices of bicritical rational maps is the existence of Tricorn-type hyperbolic components. Such a hyperbolic component is called invisible if the non-bifurcating sub-arcs on its boundary do not intersect the closure of any other hyperbolic component. Numerical evidence suggests an abundance of invisible Tricorn-type components in real slices of bicritical rational maps. In this paper, we study two different families of real bicritical maps and characterize invisible Tricorn-type components in terms of suitable topological properties in the dynamical planes of the representative maps. We use this criterion to prove the existence of infinitely many invisible Tricorn-type components in the corresponding parameter spaces. Although we write the proofs for two specific families, our methods apply to generic families of real bicritical maps.