Asymptotic escape rates and limiting distributions for multimodal maps

Mark Demers, Michael John Todd

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1 Citation (Scopus)
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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.
Original languageEnglish
Pages (from-to)1656-1705
Number of pages50
JournalErgodic Theory and Dynamical Systems
Volume41
Issue number6
Early online date9 Mar 2020
DOIs
Publication statusPublished - Jun 2021

Keywords

  • Multimodal maps
  • Hofbauer extension
  • Open systems
  • Escape rate
  • Transfer operator

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