Immediate renormalization of cubic complex polynomials with empty rational lamination

Alexander Blokh; Lex Oversteegen; Vladlen Timorin

Immediate renormalization of cubic complex polynomials with empty rational lamination

Dynamical Systems 2023-08-31 v1

Authors: Alexander Blokh , Lex Oversteegen , Vladlen Timorin

Abstract

A cubic polynomial $P$ with a non-repelling fixed point $b$ is said to be immediately renormalizable if there exists a (connected) QL invariant filled Julia set $K^*$ such that $b\in K^*$ . In that case, exactly one critical point of $P$ does not belong to $K^*$ . We show that if, in addition, the Julia set of $P$ has no (pre)periodic cutpoints, then this critical point is recurrent.

Keywords

julia set positive polynomials polynomial theory

Cite

@article{arxiv.2308.15580,
  title  = {Immediate renormalization of cubic complex polynomials with empty rational lamination},
  author = {Alexander Blokh and Lex Oversteegen and Vladlen Timorin},
  journal= {arXiv preprint arXiv:2308.15580},
  year   = {2023}
}

Comments

26 pages, 1 figure; accepted to a special issue of the MMJ dedicated to Yulij Ilyashenko's 80th Birthday. arXiv admin note: substantial text overlap with arXiv:2102.10325

Immediate renormalization of cubic complex polynomials with empty rational lamination

Abstract

Keywords

Cite

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