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Asymptotic escape rates and limiting distributions for multimodal maps

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

Published online by Cambridge University Press:  09 March 2020

MARK F. DEMERS Open the ORCID record for MARK F. DEMERS [Opens in a new window]  and
MIKE TODD Open the ORCID record for MIKE TODD [Opens in a new window]
MARK F. DEMERS
Affiliation:
Department of Mathematics, Fairfield University, Fairfield, CT 06824, USA email mdemers@fairfield.edu
MIKE TODD
Affiliation:
Mathematical Institute, University of St Andrews, North Haugh, St AndrewsKY16 9SS, UK email m.todd@st-andrews.ac.uk
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Abstract

We consider multimodal maps with holes and study the evolution of the open systems with respect to equilibrium states for both geometric and Hölder potentials. For small holes, we show that a large class of initial distributions share the same escape rate and converge to a unique absolutely continuous conditionally invariant measure; we also prove a variational principle connecting the escape rate to the pressure on the survivor set, with no conditions on the placement of the hole. Finally, introducing a weak condition on the centre of the hole, we prove scaling limits for the escape rate for holes centred at both periodic and non-periodic points, as the diameter of the hole goes to zero.

Keywords

multimodal mapsHofbauer extensionopen systemsescape ratetransfer operator

MSC classification

Primary: 37C30: Zeta functions, (Ruelle-Frobenius) transfer operators, and other functional analytic techniques in dynamical systems 37D35: Thermodynamic formalism, variational principles, equilibrium states 37E05: Maps of the interval (piecewise continuous, continuous, smooth)
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 6 , June 2021 , pp. 1656 - 1705
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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