  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.
