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High-order homoclinic tangencies of corank 2

Part of: Low-dimensional dynamical systems Smooth dynamical systems: general theory Local and nonlocal bifurcation theory

Published online by Cambridge University Press:  29 August 2025

DMITRII MINTS
DMITRII MINTS*
Affiliation:
Imperial College London , SW7 2AZ London, UK and National Research University Higher School of Economics, 25/12 Bolshaya Pecherskaya Ulitsa, 603155 Nizhny Novgorod, Russia
*
e-mail: d.mints22@imperial.ac.uk
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Abstract

We prove that in the space of $C^r$ maps $(r=2,\ldots ,\infty ,\omega )$ of a smooth manifold of dimension at least 4, there exist open regions where maps with infinitely many corank-2 homoclinic tangencies of all orders are dense. The result is applied to show the existence of maps with universal two-dimensional dynamics, that is, maps whose iterations approximate the dynamics of every map of a two-dimensional disk with an arbitrarily good accuracy. We show that maps with universal two-dimensional dynamics are $C^r$-generic in the regions under consideration.

Keywords

homoclinic tangencyNewhouse domainuniversal dynamics

MSC classification

Primary: 37C29: Homoclinic and heteroclinic orbits 37G25: Bifurcations connected with nontransversal intersection 37C20: Generic properties, structural stability 37E20: Universality, renormalization

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 46 , Issue 1 , January 2026 , pp. 218 - 262
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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