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Wandering continua, renormalization, and Lebesgue measure on Julia sets

  • Willie Rush Lim
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Abstract

We study the Lebesgue measure of wandering continua and related invariant sets in Julia sets, using the hyperbolic area method recently developed by Ye. For entire functions of finite type, we show that a wandering continuum of positive area must escape to infinity uniformly. For infinitely renormalizable unicritical polynomials, we show that almost every point in the renormalization core (the intersection of the orbits of little Julia sets) is recurrent and contained in the postcritical set. The main tool is a conservativity theorem: if f is injective on a forward invariant measurable subset E of the Julia set, then f|E is conservative and almost every point of E is recurrent and contained in the postsingular set.

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Cite

BibTeXRIS
BibTeX
@misc{lim2026wandering,
  title = {{Wandering continua, renormalization, and Lebesgue measure on Julia sets}},
  author = {Willie Rush Lim},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.05949},
  eprint = {2611.05949},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.05949},
}
RIS
TY  - UNPB
TI  - Wandering continua, renormalization, and Lebesgue measure on Julia sets
AU  - Willie Rush Lim
PY  - 2026
DA  - 2026/11/04
PB  - arXiv
DO  - 10.48550/arxiv.2611.05949
UR  - https://arxiv.org/abs/2611.05949
AB  - We study the Lebesgue measure of wandering continua and related invariant sets in Julia sets, using the hyperbolic area method recently developed by Ye. For entire functions of finite type, we show that a wandering continuum of positive area must escape to infinity uniformly. For infinitely renormalizable unicritical polynomials, we show that almost every point in the renormalization core (the intersection of the orbits of little Julia sets) is recurrent and contained in the postcritical set. The main tool is a conservativity theorem: if f is injective on a forward invariant measurable subset E of the Julia set, then f|_E is conservative and almost every point of E is recurrent and contained in the postsingular set.
ER  -