Équidistribution, comptage et approximation par irrationnels quadratiques

Parkkonen, Jouni; Paulin, Frédéric

doi:10.3934/jmd.2012.6.1

Cited by 15 publications

(46 citation statements)

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“…Many papers have given necessary and sufficient condition for when a quadratic irrational element is in the same orbit under as its Galois conjugate, see for instance [12,35,44,45] and also [42,Prop. 5.3].…”

Section: Applications To Diophantine Approximationmentioning

confidence: 99%

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Spiraling spectra of geodesic lines in negatively curved manifolds

Parkkonen

Paulin²

2010

Math. Z.

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Given a negatively curved geodesic metric space M, we study the asymptotic penetration behaviour of geodesic lines of M in small neighbourhoods of closed geodesics and of other compact convex subsets of M. We define a spiraling spectrum which gives precise information on the asymptotic spiraling lengths of geodesic lines around these objects. We prove analogs of the theorems of Dirichlet, Hall and Cusick in this context. As a consequence, we obtain Diophantine approximation results of elements of R, C or the Heisenberg group by quadratic irrational ones.

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Section: Applications To Diophantine Approximationmentioning

confidence: 99%

“…This difference is good to bear in mind when comparing our results with for example the results of [11,18,48] cited in the Introduction. We refer to [42,Lem. 5.2] for a treatment of some algebraic number theory aspects of h(α).…”

Section: Proposition 63mentioning

confidence: 99%

Spiraling spectra of geodesic lines in negatively curved manifolds

Parkkonen

Paulin²

2010

Math. Z.

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“…For every real quadratic irrational number α, Parkkonen and Paulin [9,10] introduced the quantity h(α) := 2 |α − α σ | , (1.1) which may be viewed as a measure of the complexity of α. As noted in [9], this quantity behaves in a very different way from the naïve height of α (the naïve height of an algebraic number is the maximum of the absolute values of the coefficients of its minimal defining polynomial over the rational integers), a notion which is commonly used in Diophantine approximation; see e.g.…”

Section: Introductionmentioning

confidence: 99%

“…In other words, E α 0 is the set of quadratic numbers whose continued fraction expansion is ultimately periodic with the same period as α 0 or as α σ 0 . For a real number ξ not in Q ∪ E α 0 , Parkkonen and Paulin [9,10] the approximation constant c α 0 (ξ) of ξ by elements of E α 0 and they proved that c α 0 (ξ) is always finite. Observe that it follows immediately from (1.2) that c α 0 +k (ξ + k ) = c α 0 (ξ), (1.3) for every integers k and k .…”

Section: Introductionmentioning

confidence: 99%

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On the quadratic Lagrange spectrum

Bugeaud

2013

Math. Z.

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Equidistribution and Counting Under Equilibrium States in Negative Curvature and Trees

Broise-Alamichel¹,

Parkkonen²,

Paulin³

2019

Progress in Mathematics

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In this survey based on the book by the authors [BPP], we recall the Patterson-Sullivan construction of equilibrium states for the geodesic flow on negatively curved orbifolds or tree quotients, and discuss their mixing properties, emphazising the rate of mixing for (not necessarily compact) tree quotients via coding by countable (not necessarily finite) topological shifts. We give a new construction of numerous nonuniform tree lattices such that the (discrete time) geodesic flow on the tree quotient is exponentially mixing with respect to the maximal entropy measure: we construct examples whose tree quotients have an arbitrary space of ends or an arbitrary (at most exponential) growth type. 1 1 A Patterson-Sullivan construction of equilibrium states We refer to [PPS, Chap. 3, 6, 7] and [BPP, Chap. 2, 3, 4] for details and complements on this section. Let X be (see [BPP] for a more general framework) ‚ either a complete, simply connected Riemannian manifold Ă M with dimension m at least 2 and pinched sectional curvature at most´1, ‚ or (the geometric realisation of) a simplicial tree X whose vertex degrees are uniformly bounded and at least 3. In this case, we respectively denote by EX and V X the sets of vertices and edges of X. For every edge e, we denote by opeq, tpeq, e its original vertex, terminal vertex and opposite edge. Let us fix an indifferent basepoint x˚in Ă M or in V X. Recall (see for instance [BH]) that a geodesic ray or line in X is an isometric map from r0,`8r or R respectively into X, that two geodesic rays are asymptotic if they stay at bounded distance one from the other, and that the boundary at infinity of X is the space B 8 X of asymptotic classes of geodesic rays in X endowed with the quotient topology of the compact-open topology. When X " Ă M , up to a translation factor, two asymptotic geodesic rays converge exponentially fast one to the other, and B 8 Ă M is homeomorphic to the sphere S m´1 of dimension m´1. When X is a tree, up to a translation factor, two asymptotic geodesic rays coincide after a certain time, and B 8 Ă M is homeomorphic to a Cantor set.

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Équidistribution, comptage et approximation par irrationnels quadratiques

Cited by 15 publications

References 50 publications

Spiraling spectra of geodesic lines in negatively curved manifolds

Spiraling spectra of geodesic lines in negatively curved manifolds

On the quadratic Lagrange spectrum

Equidistribution and Counting Under Equilibrium States in Negative Curvature and Trees

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