Invariant curves for endomorphisms of $${{\mathbb {P}}}^1\times {{\mathbb {P}}}^1$$

Pakovich, Fedor

doi:10.1007/s00208-021-02304-5

Math. Ann.

2022

DOI: 10.1007/s00208-021-02304-5

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Invariant curves for endomorphisms of $${{\mathbb {P}}}^1\times {{\mathbb {P}}}^1$$

Fedor Pakovich

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2024

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Cited by 2 publications

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On algebraic dependencies between Poincaré functions

PAKOVICH

2024

Ergod. Th. Dynam. Sys.

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Let A be a rational function of one complex variable of degree at least two, and $z_0$ its repelling fixed point with the multiplier $\unicode{x3bb} .$ A Poincaré function associated with $z_0$ is a function meromorphic on ${\mathbb C}$ such that , and In this paper, we study the following problem: given Poincaré functions and , find out if there is an algebraic relation between them and, if such a relation exists, describe the corresponding algebraic curve $f(x,y)=0.$ We provide a solution, which can be viewed as a refinement of the classical theorem of Ritt about commuting rational functions. We also reprove and extend previous results concerning algebraic dependencies between Böttcher functions.

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On algebraic dependencies between Poincaré functions

PAKOVICH

2024

Ergod. Th. Dynam. Sys.

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When do two rational functions have locally biholomorphic Julia sets?

Dujardin¹,

Favre

Gauthier³

2022

Trans. Amer. Math. Soc.

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In this article we address the following question, whose interest was recently renewed by problems arising in arithmetic dynamics: under which conditions does there exist a local biholomorphism between the Julia sets of two given one-dimensional rational maps? In particular we find criteria ensuring that such a local isomorphism is induced by an algebraic correspondence. This extends and unifies classical results due to Baker, Beardon, Eremenko, Levin, Przytycki and others. The proof involves entire curves and positive currents.

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Invariant curves for endomorphisms of $${{\mathbb {P}}}^1\times {{\mathbb {P}}}^1$$

Cited by 2 publications

References 24 publications

On algebraic dependencies between Poincaré functions

On algebraic dependencies between Poincaré functions

When do two rational functions have locally biholomorphic Julia sets?

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