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Multidimensional continued fractions, dynamical renormalization and KAM theory
, 2005
"... The disadvantage of ‘traditional’ multidimensional continued fraction algorithms is that it is not known whether they provide simultaneous rational approximations for generic vectors. Following ideas of Dani, Lagarias and KleinbockMargulis we describe a simple algorithm based on the dynamics of fl ..."
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Cited by 14 (5 self)
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The disadvantage of ‘traditional’ multidimensional continued fraction algorithms is that it is not known whether they provide simultaneous rational approximations for generic vectors. Following ideas of Dani, Lagarias and KleinbockMargulis we describe a simple algorithm based on the dynamics of flows on the homogeneous space SL(d, Z) \ SL(d, R) (the space of lattices of covolume one) that indeed yields best possible approximations to any irrational vector. The algorithm is ideally suited for a number of dynamical applications that involve small divisor problems. As an example, we explicitly construct a renormalization scheme for the linearization of vector fields on tori of arbitrary dimension.
SYMBOLIC DYNAMICS FOR THE MODULAR SURFACE AND BEYOND
, 2007
"... In this expository article we describe the two main methods of representing geodesics on surfaces of constant negative curvature by symbolic sequences and their development. A geometric method stems from a 1898 work of J. Hadamard and was developed by M. Morse in the 1920s. It consists of recording ..."
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Cited by 7 (0 self)
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In this expository article we describe the two main methods of representing geodesics on surfaces of constant negative curvature by symbolic sequences and their development. A geometric method stems from a 1898 work of J. Hadamard and was developed by M. Morse in the 1920s. It consists of recording the successive sides of a given fundamental region cut by the geodesic and may be applied to all finitely generated Fuchsian groups. Another method, of arithmetic nature, uses continued fraction expansions of the end points of the geodesic at infinity and is even older—it comes from the Gauss reduction theory. Introduced to dynamics by E. Artin in a 1924 paper, this method was used to exhibit dense geodesics on the modular surface. For 80 years these classical works have provided inspiration for mathematicians and a testing ground for new methods in dynamics, geometry and combinatorial group theory. We present some of the ideas, results (old and recent), and interpretations that illustrate the multiple facets of the subject.
Arithmetic coding of geodesics on the modular surface via continued fractions
 59–77, CWI Tract 135, Math. Centrum, Centrum Wisk. Inform
, 2005
"... Abstract. In this article we present three arithmetic methods for coding oriented geodesics on the modular surface using various continued fraction expansions and show that the space of admissible coding sequences for each coding is a onestep topological Markov chain with countable alphabet. We als ..."
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Cited by 6 (1 self)
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Abstract. In this article we present three arithmetic methods for coding oriented geodesics on the modular surface using various continued fraction expansions and show that the space of admissible coding sequences for each coding is a onestep topological Markov chain with countable alphabet. We also present conditions under which these arithmetic codes coincide with the geometric code obtained by recording oriented excursions into the cusp of the modular surface.
Multidimensional Euclidean algorithms, numeration and substitutions, Integers
"... Abstract: The aim of this survey is to discuss multidimensional continued fraction and Euclidean algorithms from the viewpoint of numeration systems, substitutions, and the symbolic dynamical systems they generate. We will mainly focus on two types of multidimensional algorithms, namely, unimodular ..."
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Cited by 4 (1 self)
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Abstract: The aim of this survey is to discuss multidimensional continued fraction and Euclidean algorithms from the viewpoint of numeration systems, substitutions, and the symbolic dynamical systems they generate. We will mainly focus on two types of multidimensional algorithms, namely, unimodular Markovian ones which include the most classical ones like e.g. JacobiPerron algorithm, and algorithms issued from lattice reduction. We will discuss these algorithms motivated by the study of substitutive dynamical systems, symbolic dynamical systems with low complexity function, and discrete geometry.
A TwoDimensional Minkowski?(x) Function
, 2002
"... A onetoone continuous function from a triangle to itself is defined that has both interesting number theoretic and analytic properties. This function is shown to be a natural generalization of the classical Minkowski?(x) function. It is shown there exists a natural class of pairs of cubic irration ..."
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A onetoone continuous function from a triangle to itself is defined that has both interesting number theoretic and analytic properties. This function is shown to be a natural generalization of the classical Minkowski?(x) function. It is shown there exists a natural class of pairs of cubic irrational numbers in the same cubic number field that are mapped to pairs of rational numbers, in analog to?(x) mapping quadratic irrationals on the unit interval to rational numbers on the unit interval. It is also shown that this new function satisfies an analog to the fact that?(x), while increasing and continuous, has derivative zero almost everywhere. 1
Geodesic Flow on the Quotient Space of the ⟩ on the Upper Half Plane Action of ⟨z + 2, − 1
"... Let G be the group generated by z ↦ → z +2 and z ↦ → − 1, z ∈ C. This z group acts on the upper half plane and the associated quotient surface is topologically a sphere with two cusps. Assigning a “geometric ” code to an oriented geodesic not going to cusps, with alphabets in Z \ {0}, enables us to ..."
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Let G be the group generated by z ↦ → z +2 and z ↦ → − 1, z ∈ C. This z group acts on the upper half plane and the associated quotient surface is topologically a sphere with two cusps. Assigning a “geometric ” code to an oriented geodesic not going to cusps, with alphabets in Z \ {0}, enables us to conjugate the geodesic flow on this surface to a special flow over the symbolic space of these geometric codes. We will show that for k ≥ 1, a subsystem with codes from Z \ {0, ±1, ±2, · · · , ±k} is a TBS: topologically Bernouli scheme. For similar codes for geodesic flow on modular surface, this was true for k ≥ 3. We also give bounds for the entropy of these subsystems.
20. Geometric coding 61
"... 21. Symbolic representation of geodesics via geometric code. 65 22. Arithmetic codings 68 23. Reduction theory conjecture 72 24. Symbolic representation of geodesics via arithmetic codes 75 25. Complexity of the geometric code 78 ..."
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21. Symbolic representation of geodesics via geometric code. 65 22. Arithmetic codings 68 23. Reduction theory conjecture 72 24. Symbolic representation of geodesics via arithmetic codes 75 25. Complexity of the geometric code 78
SYMBOLIC DYNAMICS FOR THE MODULAR SURFACE AND BEYOND SVETLANA KATOK AND ILIE UGARCOVICI
, 2006
"... Regarding the fundamental investigations of mathematics, there is no final ending... no first beginning. —Felix Klein All new is wellforgotten old. —A proverb Abstract. In this expository article we describe the two main methods of representing geodesics on surfaces of constant negative curvature b ..."
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Regarding the fundamental investigations of mathematics, there is no final ending... no first beginning. —Felix Klein All new is wellforgotten old. —A proverb Abstract. In this expository article we describe the two main methods of representing geodesics on surfaces of constant negative curvature by symbolic sequences and their development. A geometric method stems from a 1898 work of J. Hadamard and was developed by M. Morse in the 1920s. It consists of recording the successive sides of a given fundamental region cut by the geodesic and may be applied to all finitely generated Fuchsian groups. Another method, of arithmetic nature, uses continued fraction expansions of the end points of the geodesic at infinity and is even older—it comes from the Gauss reduction theory. Introduced to dynamics by E. Artin in a 1924 paper, this method was used to exhibit dense geodesics on the modular surface. For 80 years these classical works have provided inspiration for mathematicians and a testing ground for new methods in dynamics, geometry and combinatorial group theory. We present some of the ideas, results (old and recent), and interpretations that illustrate the multiple facets of the subject. Contents 1.