By A. Jaffe (Chief Editor)

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**Extra info for Communications in Mathematical Physics - Volume 206**

**Example text**

4. The Upper Bounds . . . . . . . . . . . . . 1 The general case . . . . . . . . . . . . . 1 . . . . . . . . . . . 3 The case when V has power law singularities . . . . . 5. Appendix . . . . . . . . . . . . . . . . 1 The structure of the Poisson potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 61 62 62 65 65 75 76 76 79 80 81 81 81 87 91 91 58 F.

Sk ) ∈ Rk+ s1 + · · · + sk ≤ t we get k Pk (t) = Pk (t) := Lk (t) i=1 e−si dsi t K −t e . K! 6 given for boxes on the simplex Lk (t). Let’s suppose that f (k, U ) < 1, otherwise there is nothing to prove. Hence the integer defined by N = [k/f (k, U )k+1 ] is bigger than k. We consider the uniform partition of [0, t]k by cubes of size t/N. Let k be the union of those cubes Qk included in the interior of Lk (t), for which for any (s1 , . . e. the union of those cubes such that there exists which gives the classical result PK (t) − PK+1 (t) = Statistics of Return Times 45 s2 t s t Fig.

If µ denotes the absolutely continuous invariant measure, we prove Poisson statistics (in the sense precise above), by giving an explicit approximation of the asymptotic law in terms of the measure of the set n , where in this case n is a decreasing sequence of cylinder sets chosen around almost all points in the interval. To be precise the error is of the type: µ( n )β , for any β < 1 − α, and therefore β is explicitly related to α and optimized just by 1 − α. For the distributions of the K th return times the bounds simply become µ( n )β/K .

### Communications in Mathematical Physics - Volume 206 by A. Jaffe (Chief Editor)

by Charles

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