Combinatorial Hubbard trees for postcritically infinite unicritical polynomials and exponential maps

Malte Hassler; Dierk Schleicher

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arXiv · 2401.10378

Combinatorial Hubbard trees for postcritically infinite unicritical polynomials and exponential maps

Malte Hassler·Dierk Schleicher

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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).

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Malte Hassler, Dierk Schleicher. 2024-01-18. Combinatorial Hubbard trees for postcritically infinite unicritical polynomials and exponential maps. https://arxiv.org/abs/2401.10378

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