Robust non-hyperbolic fibred quadratic polynomial dynamics

Igsyl Domínguez; Mario Ponce

doi:10.3934/dcds.2023080

Article Contents

2023, Volume 43, Issue 11: 4099-4112. Doi: 10.3934/dcds.2023080

This issue Previous Article Classification of solutions for some mixed order elliptic system Next Article Phase transition for the McKean-Vlasov equation of weakly coupled Hodgkin-Huxley oscillators

Robust non-hyperbolic fibred quadratic polynomial dynamics

Igsyl Domínguez^1, , and
Mario Ponce^1,

1.
Pontificia Universidad Católica de Chile, Chile

^*Corresponding author: Igsyl Domínguez
^*Corresponding author: Igsyl Domínguez

Received: June 2023

Revised: July 2023

Early access: July 31, 2023

Published online: July 31, 2023

The first author is supported by [ANID grant number 21191154].
The second author is partially supported by [ANID-DONDECYT number 1220032].

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Abstract

We consider the dynamics of fibred quadratic polynomials over an irrational rotation of the circle. We construct a simple mechanism, so-called critical connection, that implies non-hyperbolicity. To exhibit that critical connection is a robust property in the parameter space of fibred quadratic polynomials over a fixed irrational rotation of the circle. Since there exist fibred quadratic polynomials that have such phenomena, we conclude that hyperbolicity is non-dense in this sense. In order to obtain our main results, we prove that, in the hyperbolic case, the filled-in Julia set varies continuously with the fibres. Even though the existence of robust mechanisms that give rise to non-hyperbolicity is a technique that has become standard, the novelty of this work has to do with the fact that we were able to establish this mechanism in a polynomial family of degree $ 2 $ of the complex plane, fibred over a non-chaotic map.

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Mathematics Subject Classification: Primary: 37F10; Secondary: 37F15.

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