A Note On Core Decomposition Of Mandelbrot Set

Every compactum K subset of (C) over cap has a core decomposition D-K(PC) with Peano quotient, which is the finest one among all upper semi-continuous decompositions into sub-continua such that the quotient space is a Peano compactum [14]. The properties of core decomposition can provide information about the topology of K. Each member of D-K(PC) is called an atom of K. In this paper, it is shown that if {chi} is an atom of a continuum K subset of (C) over cap then K is locally connected at chi; however, the converse is not true. Moreover, we will obtain singleton atoms of the Mandelbrot set M, based on well known points at which M is locally connected. (c) 2020 Elsevier Ltd. All rights reserved.