Accessible hyperbolic components in anti-holomorphic dynamics

TL;DR

This paper investigates the accessibility of hyperbolic components in the tricorn, revealing infinitely many that are accessible despite the complex boundary structure and prior expectations of inaccessibility.

Contribution

The authors demonstrate the existence of infinitely many accessible hyperbolic components in the tricorn, challenging previous assumptions about their inaccessibility at large odd periods.

Findings

The boundary of hyperbolic components of odd period contains inaccessible arcs.

As period increases, decorations become more complex.

Contrary to expectations, infinitely many hyperbolic components are accessible.

Abstract

The tricorn, the connectedness locus of the anti-holomorphic quadratic family, is known to be non-locally connected. The boundary of every hyperbolic component of odd period contains arcs that are inaccessible from the complement of the tricorn. As the period increases, the decorations become more and more complicated, and it seems natural to think that every hyperbolic component of sufficiently large and odd period is inaccessible. Contrary to this expectation, we show that the tricorn contains infinitely many hyperbolic components that are accessible from the complement.