- Title Pages
- Preface
- Introduction
- Arithmetic of Unicritical Polynomial Maps
- Les racines des composantes hyperboliques de M sont des quarts d'entiers algébriques
- Dynamical cores of topological polynomials
- The quadratic dynatomic curves are smooth and irreducible
- Multicorns are not path connected
- Leading monomials of escape regions
- Limiting behavior of Julia sets of singularly perturbed rational maps
- On (non-)local connectivity of some Julia sets
- Perturbations of weakly expanding critical orbits
- Unmating of rational maps: Sufficient criteria and examples
- A framework toward understanding the characterization of holomorphic dynamics
- Metric stability for random walks (with applications in renormalization theory)
- Milnor's conjecture on monotonicity of topological entropy: Results and questions
- Entropy in dimension one
- On Ecalle-Hakim 's theorems in holomorphic dynamics
- Index theorems for meromorphic self-maps of the projective space
- Dynamics of automorphisms of compact complex surfaces
- Bifurcation currents and equidistribution in parameter space
- Entropy for hyperbolic Riemann surface laminations I
- Entropy for hyperbolic Riemann surface laminations II
- Intersection theory for ergodic solenoids
- Invariants of four-manifolds with flows via cohomological field theory
- Two papers which changed my life: Milnor's seminal work on flat manifolds and bundles
- Mil nor's problem on the growth of groups and its consequences
- Contributors
- Index

# 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

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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- Title Pages
- Preface
- Introduction
- Arithmetic of Unicritical Polynomial Maps
- Les racines des composantes hyperboliques de M sont des quarts d'entiers algébriques
- Dynamical cores of topological polynomials
- The quadratic dynatomic curves are smooth and irreducible
- Multicorns are not path connected
- Leading monomials of escape regions
- Limiting behavior of Julia sets of singularly perturbed rational maps
- On (non-)local connectivity of some Julia sets
- Perturbations of weakly expanding critical orbits
- Unmating of rational maps: Sufficient criteria and examples
- A framework toward understanding the characterization of holomorphic dynamics
- Metric stability for random walks (with applications in renormalization theory)
- Milnor's conjecture on monotonicity of topological entropy: Results and questions
- Entropy in dimension one
- On Ecalle-Hakim 's theorems in holomorphic dynamics
- Index theorems for meromorphic self-maps of the projective space
- Dynamics of automorphisms of compact complex surfaces
- Bifurcation currents and equidistribution in parameter space
- Entropy for hyperbolic Riemann surface laminations I
- Entropy for hyperbolic Riemann surface laminations II
- Intersection theory for ergodic solenoids
- Invariants of four-manifolds with flows via cohomological field theory
- Two papers which changed my life: Milnor's seminal work on flat manifolds and bundles
- Mil nor's problem on the growth of groups and its consequences
- Contributors
- Index