## Dynamical Numbers: Interplay Between Dynamical Systems and by Sergiy Kolyada, Yuri Manin, Martin Moller, Pieter Moree,

By Sergiy Kolyada, Yuri Manin, Martin Moller, Pieter Moree, Thomas Ward

Includes the court cases of the task 'Dynamical Numbers: interaction among Dynamical platforms and quantity thought' held on the Max Planck Institute for arithmetic (MPIM) in Bonn, from 1 could to 31 July, 2009, and the convention of an analogous name, additionally held on the Max Planck Institute, from 20 to 24 July, 2009--Preface, p. vii

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20, (1976), 555-588. il Contemporary Mathematics Volume 532, 2010 Geodesic ﬂow on the Teichm¨ uller disk of the regular octagon, cutting sequences and octagon continued fractions maps John Smillie and Corinna Ulcigrai Abstract. In this paper we give a geometric interpretation of the renormalization algorithm and of the continued fraction map that we introduced in [15] to give a characterization of symbolic sequences for linear ﬂows in the regular octagon. We interpret this algorithm as renormalization on the Teichm¨ uller disk of the octagon and explain the relation with Teichm¨ uller geodesic ﬂow.

2. (A correspondence principle) Let G be a locally compact group. Given a nonempty subset L ⊂ G, there exists a compact metric G-space X and an open subset A ⊂ X such that g1−1 A ∩ g2−1 A ∩ · · · ∩ gk−1 A = ∅ =⇒ g1 a, . . , gk a ∈ L , for some a ∈ G. If, moreover, m is a probability measure on G and ρ an mstationary mean on G with ρ(L) > 0, then there exists an m-stationary probability measure μ on X with μ(A) ≥ ρ(L). Proof. Let f : G → [0, 1] be a left uniformly continuous function such that f (g) = 1 for every g ∈ L /2 and f (g) = 0 for every g ∈ L .

2. 2000 Mathematics Subject Classiﬁcation. Primary 37B10, 11J70, 37E35, 37C40. Partially supported by NSF Grant DMS-0901521. Partially supported by an RCUK Academic Fellowship. c 0000 (copyright Society holder) c 2010 American Mathematical 1 29 30 2 JOHN SMILLIE AND CORINNA ULCIGRAI D 8 6 A 1 4 3 B 2 5 5 C C 7 7 3 B 2 8 1 4 6 D A Figure 1. A linear trajectory in the octagon. Outline. In the remainder of this section we give some basic deﬁnitions and we give a brief exposition of the renormalization schemes and of the continued fraction algorithm introduced in [15].