Asymptotic escape rates and limiting distributions for multimodal maps

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 nonperiodic points, as the diameter of the hole goes to zero.

Escape rate and conditional escape rate from a probabilistic point of view

“…Let ϕ : M → R ∪ {+∞} be a continuous function achieving its maximum on a measure zero set Λ. Let {u n } be a non-decreasing sequence of real numbers with u n ր sup ϕ, such that the open sets U n defined by (5) satisfy Assumption 1 for a sequence r n that decreases to 0 as n → ∞. Let κ n be the smallest positive integer for which diam A κn ≤ r n and assume that:…”

“…In this case a dichotomy has been established for many systems which shows the local escape rate to be equal to one at non-periodic points and equal to the extremal index at periodic points. See the classical work [9] for conformal attractors, [2,5] for the transfer operator approach for interval maps, and [15] for a probabilistic approach which applies to systems in higher dimension. This mirrors the behaviour of the limiting return times distributions that are Poisson at non-periodic points, and Pólya-Aeppli compound Poisson at periodic points in which case the compounded geometric distribution has the weights given by the extremal index θ ∈ (0, 1).…”