Peano Model for Planar Compacta and a Lemma by Beardon

TL;DR

This paper studies the core decomposition of planar compact sets with Peano hyperspaces, showing its invariance under embedding and extending results related to rational maps and the Mandelbrot set.

Contribution

It introduces a new topologically invariant lambda function and extends Beardon and Curry's results to broader classes of sets and maps.

Findings

Core decomposition is embedding-independent.

Pre-image components belong to the core decomposition of the pre-image set.

Lambda function is topologically invariant and relates to the Mandelbrot set.

Abstract

It is known that, among all the monotone decompositions of a planar compact set K with Peano hyperspaces, there exists a unique one that is finer than all the others. We call it the "core decomposition" of K with Peano hyperspace. The resulted hyperspace under quotient topology will be referred to as the "Peano model" for K. We show that the core decomposition is independent of the embedding of K into the plane. Given a rational function f with degree at least 2 that is independent of K. A well known result by Beardon says that the pre-image for any element d of the core decomposition has finitely many components, each of which is mapped by f onto d. We show that those components belong to the core decomposition of L, the pre-image of K under the above mentioned rational map f. This provides an affirmative answer to Question 5.4 proposed by Curry (MR2642461) and extends earlier partial results by Blokh-Curry-Oversteegen (MR2737795 and MR3008890), when K is assumed to be unshielded and the rational map f is assumed to be a polynomial, under which K is completely invariant. The previous result is also connected with a well known Factor Theorem developed by Whyburn. We also introduce a lambda function from K to the set of non-negative integers, such that this function is constantly zero if and only if K is a Peano space. This function and its maximum are topologically invariant, while its level set at zero is of particular interest. For instance, when K is the Mandelbrot set we find close relations between the level set at zero and the hyperbolic components. Further discussions on the lambda function can be expected.