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Mirit Lewin. Auteur du texte 44 Scot Adams. Ethan Akin. Algebraic and topological dynamics ; Bonn, Allemagne. Lluis Alseda.

We use notations introduced in Chap. The following proposition, which is analogous to Proposition III. Prove Proposition 2. Hint: use Exercises 2.

This result, added to Proposition 2. This implies, by Proposition 2. The preceding argument, together with the Lorentzian characterization of conical points Proposition 1. One can deduce from this theorem and from Corollary I. II then, by Proposition 1. As we have shown in Chaps. However, one can state the following properties which follow directly from Theorems III. The Lorentzian model that we propose below brings the methods that were used in Chap.

V into play. For this reason, many of the proofs are left as exercises.

Hint: see Exercises V. Hint: Rewrite the proof of Proposition V. This result, together with Proposition 1. Hedlund [37] and L. Greenberg [4], the study of orbits of groups acting linearly on a vector space has become a research area in its own right. This result has since been extended by J. Conze and Y. In this chapter and Sect. In the metric context, there are many applications of this new point of view [5, 32] and [60, Chap.

In the second step, we restrict our attention to the modular group and rediscover, in the spirit of Chap. III, some classical results of the theory of Diophantine approximations. As we will see, the answer depends on properties of the point x. Thus one obtains: Proposition 1. Furthermore, by Theorem III. According to Proposition 1.

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As consequence, we obtain Corollary 1. Actually, by Corollary 1. Hint: see Property III. This is not the case if x is conical. This shows that t1 belongs to E [z, x. Fix a real number t in E [z, x. Using Property I. In conclusion, the upper bound of this set is at least that of E [z, x. Is the converse true? In the following section, we give an answer to this question. It remains to consider case ii.

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Thus one obtains the following characterization: Proposition 2. II Property II. Recall from Proposition II. The following proposition characterizes the geometrically badly approximated points in coding terms.

Let x be a conical point in L S p, h. In this case, by Property II. We begin with a discussion of three well-known results from number theory. We will prove them in this section using a hyperbolic point of view. One example is Proposition II. Another branch focuses on the speed of convergence of the sequence of rational numbers associated with a continued fraction expansion.

One of its classical proofs relies on some properties of the continued fraction expansion [52, Chap. The following theorem is more precise. It can be proved, for example by associating a sequence of circles to the sequence of rational numbers given by the continued fraction expansion, and by studying their relative positions [52, Chap. For example, this is the case for 2. A proof of this theorem is given, for example, in [24, Theorem 2. Our purpose here is not to gain simplicity but to illustrate the fact that the mathematical world is not compartmentalized.

Recall from Lemma I. We will essentially recycle the arguments used in the proof of Proposition I.

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We are now ready to prove Theorems 3. Proof of Theorem 3. As shown in Chap. II Exercise II. By Lemma 3. In other words, on H one has the expression Fig. At the end of Sect. We prove now the reverse inequality. Thus without loss of generality one can assume that x is in [0, 1]. Denoted by [0; n1 ,.

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As in Sect. Recall the following facts from Exercise II. In the second case, x is related to the golden ratio. It follows from the proof of Theorem 3. It now remains only to prove Theorem 3. Thus x is badly approximately if and only if x is geometrically badly approximated. Recall that, by Corollary 2. By Lemmas 3. It follows that this ray intersects at least k vertical geodesics of the form Fig. Returning to the geometric construction of the continued fraction expansion discussed in Sect. Some years after, the key idea to built a relationship between fractions and circles appeared in an elementary and informative paper of Ford [30].

The geometric approach to numbers, as presented in Sect. II [15, 35, 56]. It also allows some questions of number theory to be formulated in terms of dynamics. The S. The metric theory of approximations can likewise be approached from this geometric angle, and can be generalized by allowing the role of the Lebesgue 4 Comments measure to be played by a Patterson measure [51, 62].

Due to the initiative of M. Raghunathan, this approach has been similarly developed by G. Margulis, S. Dani and many other mathematicians to solve some problems in Diophantine approximation in Rn [60, Chap. IV] and [46].