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PreprintCC BY 4.0arXiv

Spectrum of the basilica group and the associated real and complex 2D dynamics

  • Eric Bedford
  • Rostislav Grigorchuk
  • Nguyen-Bac Dang
  • Mikhail Lyubich
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Abstract

In this paper, we study the iteration over the real and complex loci of a specific rational map, found by Grigorchuk and Zuk. This two dimensional map arises as the renormalization map which yields the density of states measure associated with the iterated monodromy group of the Basilica Julia set. It admits two natural complex currents, and we construct their real analogs and show that they are laminar. Finally, this yields some consequences on the properties of the density of states measures and its support.

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Cite

BibTeXRIS
BibTeX
@misc{bedford2026spectrum,
  title = {{Spectrum of the basilica group and the associated real and complex 2D dynamics}},
  author = {Eric Bedford and Rostislav Grigorchuk and Nguyen-Bac Dang and Mikhail Lyubich},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.31961},
  eprint = {2610.31961},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.31961},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Spectrum of the basilica group and the associated real and complex 2D dynamics
AU  - Eric Bedford
AU  - Rostislav Grigorchuk
AU  - Nguyen-Bac Dang
AU  - Mikhail Lyubich
PY  - 2026
DA  - 2026/10/22
PB  - arXiv
DO  - 10.48550/arxiv.2610.31961
UR  - https://arxiv.org/abs/2610.31961
AB  - In this paper, we study the iteration over the real and complex loci of a specific rational map, found by Grigorchuk and Zuk. This two dimensional map arises as the renormalization map which yields the density of states measure associated with the iterated monodromy group of the Basilica Julia set. It admits two natural complex currents, and we construct their real analogs and show that they are laminar. Finally, this yields some consequences on the properties of the density of states measures and its support.
ER  -