Number Theory

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

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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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Additional resources for Dynamical Numbers: Interplay Between Dynamical Systems and Number Theory, A Special Program May 1-July 31, 2009, International Conference July 20-24, 2009, Max Planack I

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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].

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