Asymptotic escape rates and limiting distributions for multimodal maps. (English) Zbl 1472.37024
Summary: 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.
MSC:
| 37C30 | Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc. |
| 37C40 | Smooth ergodic theory, invariant measures for smooth dynamical systems |
| 37D35 | Thermodynamic formalism, variational principles, equilibrium states for dynamical systems |
| 37E05 | Dynamical systems involving maps of the interval |
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