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Frontiers in Complex DynamicsIn Celebration of John Milnor's 80th Birthday$
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Araceli Bonifant, Misha Lyubich, and Scott Sutherland

Print publication date: 2014

Print ISBN-13: 9780691159294

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691159294.001.0001

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Dynamical cores of topological polynomials

Dynamical cores of topological polynomials

Chapter:
(p.27) Dynamical cores of topological polynomials
Source:
Frontiers in Complex Dynamics
Author(s):

Alexander Blokh

Lex Oversteegen

Ross Ptacek

Vladlen Timorin

, Araceli Bonifant, Mikhail Lyubich, Scott Sutherland
Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691159294.003.0004

This chapter defines the (dynamical) core of a topological polynomial (and the associated lamination). This notion extends that of the core of a unimodal interval map. Two explicit descriptions of the core are given: one related to periodic objects and one related to critical objects. Topological polynomials are topological dynamical systems that generalize complex polynomials with locally connected Julia sets restricted to their Julia sets and considered up to topological conjugacy. This chapter aims to illustrate the analogy between the dynamics of topological polynomials on their cutpoints and cut-atoms and interval dynamics. For example, it is known that for interval maps, periodic points and critical points play a significant, if not decisive, role. This chapter thus attempts to establish similar facts for topological polynomials.

Keywords:   dynamical cores, topological polynomials, unimodal interval map, periodic objects, critical objects, interval dynamics

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