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Stable intersections of conformal Cantor sets

Part of: Local and nonlocal bifurcation theory Complex dynamical systems Classical measure theory

Published online by Cambridge University Press:  25 October 2021

HUGO ARAÚJO  and
CARLOS GUSTAVO MOREIRA
HUGO ARAÚJO*
Affiliation:
Departamento de Ciências Exatas e Aplicadas, Universidade Federal de Ouro Preto, João Monlevade 35931-008, Minas Gerais, Brazil
CARLOS GUSTAVO MOREIRA
Affiliation:
Instituto de Matemática Pura e Aplicada, Rio de Janeiro 22460-320, Rio de Janeiro, Brazil (e-mail: gugu@impa.br)
*
e-mail: hugo.araujo@ufop.edu.br
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Abstract

We investigate stable intersections of conformal Cantor sets and their consequences to dynamical systems. First we define this type of Cantor set and relate it to horseshoes appearing in automorphisms of $\mathbb {C}^2$. Then we study limit geometries, that is, objects related to the asymptotic shape of the Cantor sets, to obtain a criterion that guarantees stable intersection between some configurations. Finally, we show that the Buzzard construction of a Newhouse region on $\mathrm{Aut}(\mathbb {C}^2)$ can be seen as a case of stable intersection of Cantor sets in our sense and give some (not optimal) estimate on how ‘thick’ those sets have to be.

Keywords

Cantor setssmooth dynamicsbifurcation theorysymbolic dynamics

MSC classification

Primary: 28A80: Fractals 37G25: Bifurcations connected with nontransversal intersection
Secondary: 37F99: None of the above, but in this section

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 43 , Issue 1 , January 2023 , pp. 1 - 49
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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