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...rding to the standard lexicographical order ωi = êi1 ∧êi2 ∧ · · · ∧êik , 1 ≤ i1 < i2 < · · · < ik ≤ M, i = 1, 2, · · ·,dk, (24) any k–vector ū ∈ Λ k (V ) can be represented as dk ∑ ū = ūiωi , ūi ∈ R. =-=(25)-=- i=1 A k–vector which can be written as the wedge product of k linear independent vectors of V is called decomposable. Of course, if the k vectors are linearly dependent we get the zero k–vector (22)....

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...arily short unstable manifolds are dense in M , our foliation is ‘holey’ (it covers a Cantor-like set), there infinitely many gaps in Mm where unstable manifolds fail to reach one of the two long sides of Mm. We need to estimate the relative measure of gaps in Mm. For x ∈ M , let ru(x) denote the distance from x to the nearer endpoint of the unstable manifold passing through x (if none exists, we put ru(x) = 0). Then for any stable curve W ⊂ M and ε > 0 we have m(x ∈ W : ru(x) < ε) < Cε where C > 0 is a constant (independent of W ) and m denotes the Lebesgue measure on W . (For the proof, see [12]: it is shown in [12, Section 5.12] that this estimate follows from the so-called first growth lemma in the case of dispersing billiards, and the argument applies to stadia without change; and the first growth lemma for stadia is proved in [12, Section 8.14].) Thus the union of all gaps has relative measure O(1/m) in Mm, so it can be simply excluded from Mm. After all bad parts of Mm are removed, as described above, we obtain the desired subset Mm ⊂ Mm. The conditional measure on each unstable manifold W is smooth and its density ρ satisfies ∣ ∣ ∣ d dx ln ρ(x) ∣ ∣ ∣ ≤ C|W |1/2 , where C > 0 i...

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...timate with uniform rate function I for all (not necessarily invariant) probability measures ν ∈ B. Assume that µ(g) = 0, let ν ∈ B be a probability measure and distribute (g ◦ T j)j∈N according to ν. Then, (b) (g, T, ν) satisfies the Central Limit Theorem; (c) the process (g ◦ T j)j∈N satisfies an almost-sure invariance principle. The proof of this theorem appears in Section 6. 2.5 Known Facts about the Lorentz gas Before exploring the properties of the Banach spaces we have defined, we recall some of the important properties of dispersing billiards that we shall need and refer the reader to [BSC1, BSC2, CM] for details. 2.5.1 Hyperbolicity Since we have assumed that the scatterers have strictly positive curvature K(x) at each x ∈ M , there exist constants Kmin,Kmax, τmin such that 0 < Kmin ≤ K(x) ≤ Kmax, τmin ≤ τ(x), ∀x ∈M. This allows us to define global stable and unstable cones as follows. Let (dr, dϕ) be an element of the tangent space. Then Cu(x) := {(dr, dϕ) ∈ TxM : Kmin ≤ dϕ dr ≤ Kmax + 1 τmin } and Cs(x) := {(dr, dϕ) ∈ TxM : −Kmax − 1 τmin ≤ dϕ dr ≤ −Kmin}. Note that the angle between Cu(x) and Cs(x) is uniformly bounded away from zero. The cones also enjoy the following two properties. ...

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...ts fractional powers. The results are conditional on a number of conjectures, and are corroborated by numerical simulations in up to ten dimensions. 1 Introduction The Lorentz gas is a billiard model =-=[13]-=-, in which a point particle moves in straight lines with unit speed except for specular collisions with hard scatterers, most often disks or balls. It is an “extended” billiard in that it moves in an ...

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...of adjacent arcs intersects only at the common endpoint. The resulting billiard is semidispersing, thus belongs to the standard class and is well-know to be uniformly hyperbolic, Bernoulli, and so on =-=[CM]-=-. Now perturb the fourth side into a focusing circular arc of curvature kf ≪ 1. Now matter how small the perturbation, this new billiard will never satisfy Wojtkowski’s principle and is not currently ...