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Abstract: For a fixed initial reference measure, we study the dependence of the escape rate on the hole for a smooth or piecewise smooth hyperbolic map. First, we prove the existence and Holder continuity of the escape rate for systems with small holes admitting Young towers. Then we consider general holes for Anosov diffeomorphisms, without size or Markovian restrictions. We prove bounds on the upper and lower escape rates using the notion of pressure on the survivor set and show that a variational principle holds under generic conditions. However, we also show that the escape rate function forms a devil's staircase with jumps along sequences of regular holes and present examples to elucidate some of the difficulties involved in formulating a general theory.
Abstract: We study the relation between escape rates and pressure in general dynamical systems with holes, where pressure is defined to be the difference between entropy and the sum of positive Lyapunov exponents. Central to the discussion is the formulation of a class of invariant measures supported on the survivor set over which we take the supremum to measure the pressure. Upper bounds for escape rates are proved for general diffeomorphisms of manifolds, possibly with singularities, for arbitrary holes and natural initial distributions including Lebesgue and SRB measures. Lower bounds do not hold in such generality, but for systems admitting Markov tower extensions with spectral gaps, we prove the equality of the escape rate with the absolute value of the pressure and the existence of an invariant measure realizing the escape rate, i.e. we prove a full variational principle. As an application of our results, we prove a variational principle for the billiard map associated with a planar Lorentz gas of finite horizon with holes.
Abstract: Consider a piecewise smooth expanding map of the interval possessing several invariant intervals and the same number of ergodic absolutely continuous invariant probability measures (ACIMs). After this system is perturbed to make the invariant sets merge in such a way that there is a unique ACIM, we show how to approximate the diffusion coefficient for an observable of bounded variation by the diffusion coefficient of a related continuous time Markov chain.
Abstract: We consider a piecewise smooth expanding map on an interval which has two invariant subsets of positive Lebesgue measure and exactly two ergodic absolutely continuous invariant probability measures (ACIMs). When this system is perturbed slightly to make the invariant sets merge, we describe how the unique ACIM of the perturbed map can be approximated by a convex combination of the two initial ergodic ACIMs. The result is generalized to the case of finitely many invariant components.
Abstract: We study the billiard map corresponding to a periodic Lorentz gas in 2-dimensions in the presence of small holes in the table. We allow holes in the form of open sets away from the scatterers as well as segments on the boundaries of the scatterers. For a large class of smooth initial distributions, we establish the existence of a common escape rate and normalized limiting distribution. This limiting distribution is conditionally invariant and is the natural analogue of the SRB measure of a closed system. Finally, we prove that as the size of the hole tends to zero, the limiting distribution converges to the smooth invariant measure of the billiard map.
Abstract: We study a heavy piston of mass M that separates finitely many ideal, unit mass gas particles moving in two or three dimensions. Neishtadt and Sinai previously determined a method for finding this system's averaged equation and showed that its solutions oscillate periodically. Using averaging techniques, we prove that the actual motions of the piston converge in probability to the predicted averaged behavior on the time scale M^ {1/2} when M tends to infinity while the total energy of the system is bounded and the number of gas particles is fixed.
Abstract: We study a heavy piston that separates finitely many ideal gas particles moving inside a one-dimensional gas chamber. Using averaging techniques, we prove precise rates of convergence of the actual motions of the piston to its averaged behavior. The convergence is uniform over all initial conditions in a compact set. The results extend earlier work by Sinai and Neishtadt, who determined that the averaged behavior is periodic oscillation. In addition, we investigate the piston system when the particle interactions have been smoothed. The convergence to the averaged behavior again takes place uniformly, both over initial conditions and over the amount of smoothing.
Abstract: The Darboux–Christoffel formula is a closed-form expression for the kernel of the operator that projects onto the first N of a system of one-dimensional polynomials, orthonormal with respect to some weighting function. It is a key element in the theory of Gaussian integration and in the theory of discrete variable representation or Lagrangian mesh methods for diagonalizing quantum Hamiltonians of a few degrees of freedom. The one-dimensional Darboux–Christoffel formula turns out to have a generalization that is valid in a semiclassical or asymptotic sense for a wider class of orthonormal functions than orthonormal polynomials. This class consists of the bound eigenfunctions of one-dimensional Hamiltonians with time-reversal invariance, such as kinetic-plus-potential Hamiltonians. It also has certain generalizations involving the unbound eigenfunctions of such Hamiltonians.
I mean the word proof not in the sense
of the lawyers, who set two half proofs equal to a whole one, but in
the sense of a mathematician, where half proof = 0, and it is
demanded for proof that every doubt becomes impossible.
--Carl Friedrich Gauss, quoted in G. Simmon's Calculus Gems (New York 1992).
The purpose of models is not to fit the data but to sharpen the
questions.
--Samuel Karlin, 11th R A Fisher Memorial Lecture, Royal Society April 1983.