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Combinatorial Hubbard trees for postcritically infinite unicritical polynomials and exponential maps

Malte Hassler, Dierk Schleicher

Abstract

We construct combinatorial Hubbard trees for all unicritical polynomials, and for all exponential maps, for which the critical (singular) value does not escape. More precisely, out of an external angle, or more generally a kneading sequence, we construct a forward invariant tree for which the critical orbit has the given kneading sequence. When the critical orbit of a unicritical polynomial is periodic or preperiodic, then the existence of a Hubbard tree is classical. Our trees exist even when the critical orbit is infinite (in many cases, this yields infinite trees), and even when the polynomial Julia set fails to be path connected. In the latter case, our trees cannot be reconstructed from the Julia set in complex dynamics. Our trees also exist for kneading sequences that do not arise in complex dynamics (for non complex-admissible kneading sequences).

Combinatorial Hubbard trees for postcritically infinite unicritical polynomials and exponential maps

Abstract

We construct combinatorial Hubbard trees for all unicritical polynomials, and for all exponential maps, for which the critical (singular) value does not escape. More precisely, out of an external angle, or more generally a kneading sequence, we construct a forward invariant tree for which the critical orbit has the given kneading sequence. When the critical orbit of a unicritical polynomial is periodic or preperiodic, then the existence of a Hubbard tree is classical. Our trees exist even when the critical orbit is infinite (in many cases, this yields infinite trees), and even when the polynomial Julia set fails to be path connected. In the latter case, our trees cannot be reconstructed from the Julia set in complex dynamics. Our trees also exist for kneading sequences that do not arise in complex dynamics (for non complex-admissible kneading sequences).
Paper Structure (15 sections, 26 theorems, 11 equations, 4 figures)

This paper contains 15 sections, 26 theorems, 11 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Kneading sequences and the critical path
  3. Abstract kneading sequences
  4. Precritical points on the critical path
  5. Limit points and Fatou intervals
  6. Illustrating examples
  7. The critical path and its properties
  8. The abstract Hubbard tree
  9. Constructing the Hubbard tree
  10. The structure of Hubbard trees
  11. Characteristic points and types of branch points
  12. Consistency with the classical definition
  13. Properties and examples
  14. Classification of bifurcations
  15. A trip to the zoo (i.e. to the woods): examples of trees

Key Result

Lemma 2.3

For non-periodic $\nu$ the map $\sigma: X_\nu \mapsto X_\nu$ is continuous.

Figures (4)

  • Figure 1: The critical paths of four $\star$-periodic kneading sequences. Prominent Fatou intervals are indicated by dotted lines.
  • Figure 2: The Hubbard trees of the examples considered in Figure \ref{['Fig:crit_paths']}. Dotted lines represent major Fatou intervals, vertical bars the orbits of a central itinerary and $-\alpha=\mathtt{0}\overline{\mathtt{1}}$.
  • Figure 3: Top left: A Julia set with kneading sequence $\nu=\overline{\mathtt{1}\mathtt{1}\mathtt{0}\star}$ with the critical orbit highlighted. Top right: The Julia set of its bifurcation $\tilde{\nu}=\overline{\mathtt{1}\mathtt{1}\mathtt{0}\mathtt{0}\;\mathtt{1}\mathtt{1}\mathtt{0}\star}$. Bottom left: The (abstract) Hubbard tree of $\nu$. Bottom right: The Hubbard tree of $\tilde{\nu}$. Several prominent Fatou intervals are represented by dotted lines, bars between them show the orbit of the central itinerary $\overline{\mathtt{1}\mathtt{1}\mathtt{0}\mathtt{0}}$.
  • Figure 4: Sketch of an infinite tree with non-recurrent kneading sequence $\nu=\mathtt{1}\,\mathtt{1}\mathtt{0}\,\mathtt{1}\mathtt{0}\mathtt{0}\,\mathtt{1}\mathtt{0}\mathtt{0}\mathtt{0}\,\mathtt{1}\mathtt{0}\mathtt{0}\mathtt{0}\mathtt{0}\,\mathtt{1}...$ . The fixed point $\beta=\overline{\mathtt{0}}$ is a limit point of the critical orbit, but does not belong to the tree.

Theorems & Definitions (70)

  • Definition 2.1: Abstract kneading sequence
  • Definition 2.2: Dynamical system associated to kneading sequence
  • Lemma 2.3
  • proof
  • Definition 2.4: Precritical points on critical path
  • Definition 2.5: Bifurcating $\star$-periodic kneading sequences
  • Lemma 2.6: All precritical points different
  • proof
  • Lemma 2.7: Limit points of $P_\infty$
  • proof
  • ...and 60 more