Asymptotic escape rates and limiting distributions for multimodal maps

Mark Demers, Michael John Todd

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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
Issue number6
Early online date9 Mar 2020
Publication statusPublished - Jun 2021


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


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