Complex maps without invariant densities

Henk Bruin; Mike Todd

doi:10.1088/0951-7715/19/12/012

Complex maps without invariant densities

Henk Bruin^*, Mike Todd

^*Corresponding author for this work

Pure Mathematics

Research output: Contribution to journal › Article › peer-review

5 Citations (Scopus)

Abstract

We consider complex polynomials f(z) = z^ℓ+c₁ for ℓ ∈ 2ℕ and c₁ ∈ ℝ and find some combinatorial types and values of ℓ such that there is no invariant probability measure equivalent to conformal measure on the Julia set. This holds for particular Fibonacci-like and Feigenbaum combinatorial types when ℓ is sufficiently large and also for a class of 'long-branched' maps of any critical order.

Original language	English
Article number	012
Journal	Nonlinearity
Volume	19
Issue number	12
DOIs	https://doi.org/10.1088/0951-7715/19/12/012
Publication status	Published - 1 Dec 2006

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10.1088/0951-7715/19/12/012

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title = "Complex maps without invariant densities",

abstract = "We consider complex polynomials f(z) = zℓ+c1 for ℓ ∈ 2ℕ and c1 ∈ ℝ and find some combinatorial types and values of ℓ such that there is no invariant probability measure equivalent to conformal measure on the Julia set. This holds for particular Fibonacci-like and Feigenbaum combinatorial types when ℓ is sufficiently large and also for a class of 'long-branched' maps of any critical order.",

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AB - We consider complex polynomials f(z) = zℓ+c1 for ℓ ∈ 2ℕ and c1 ∈ ℝ and find some combinatorial types and values of ℓ such that there is no invariant probability measure equivalent to conformal measure on the Julia set. This holds for particular Fibonacci-like and Feigenbaum combinatorial types when ℓ is sufficiently large and also for a class of 'long-branched' maps of any critical order.

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DO - 10.1088/0951-7715/19/12/012

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SN - 0951-7715

VL - 19

JO - Nonlinearity

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Complex maps without invariant densities

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