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Minimal strong foliations in skew-products of iterated function systems

Part of: Dynamical systems with hyperbolic behavior Smooth dynamical systems: general theory

Published online by Cambridge University Press:  03 December 2024

PABLO G. BARRIENTOS Open the ORCID record for PABLO G. BARRIENTOS [Opens in a new window]  and
JOEL ANGEL CISNEROS
PABLO G. BARRIENTOS*
Affiliation:
Instituto de Matemática e Estatística, UFF, Rua Mário Santos Braga s/n - Campus Valonguinhos, Niterói, Brazil (e-mail: joelangel@id.uff.br)
JOEL ANGEL CISNEROS
Affiliation:
Instituto de Matemática e Estatística, UFF, Rua Mário Santos Braga s/n - Campus Valonguinhos, Niterói, Brazil (e-mail: joelangel@id.uff.br)
*
e-mail: pgbarrientos@id.uff.br
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Abstract

We study locally constant skew-product maps over full shifts of finite symbols with arbitrary compact metric spaces as fiber spaces. We introduce a new criterion to determine the density of leaves of the strong unstable (and strong stable) foliation, that is, for its minimality. When the fiber space is a circle, we show that both strong foliations are minimal for an open and dense set of robustly transitive skew-products. We provide examples where either one foliation is minimal or neither is minimal. Our approach involves investigating the dynamics of the associated iterated function system (IFS). We establish the asymptotic stability of the phase space of the IFS when it is a strict attractor of the system. We also show that any transitive IFS consisting of circle diffeomorphisms that preserve orientation can be approximated by a robust forward and backward minimal, expanding, and ergodic (with respect to Lebesgue) IFS. Lastly, we provide examples of smooth robustly transitive IFSs where either the forward or the backward minimal fails, or both.

Keywords

minimality of strong foliations for skew-productsminimallity of iterated function systemsrobust transitivitystability of strict attractors

MSC classification

Primary: 37C70: Attractors and repellers, topological structure 37D10: Invariant manifold theory
Secondary: 37D30: Partially hyperbolic systems and dominated splittings

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 45 , Issue 8 , August 2025 , pp. 2273 - 2313
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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