Dynamique sur le rayon modulaire et fractions continues en caractéristique <i>p</i>

Broise-Alamichel, Anne; Paulin, Frédéric

doi:10.1112/jlms/jdm063

Cited by 10 publications

(7 citation statements)

References 10 publications

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“…Then by [BP,Theo. 3.7] and Theorem 3 (2) and (5), the map Ξ : C → (R − F q ) Z defined by x → (a i ) i∈Z is a homeomorphism such that the following diagram commutes…”

Section: Letmentioning

confidence: 95%

“…We will in particular prove in the next result that C is a natural cross-section for the right action of A : Every orbit of A in X meets C in an infinite countable subset. 3 As in [BP,Theo. 3.7…”

Section: Letmentioning

confidence: 99%

“…By well-known arguments of for instance [BuM], [Rob], [BP,§4] and [BPP,§4.4], since Γ is a lattice in G, the measure m BM is invariant on the left by the action of Γ and on the right by the geodesic flow, and defines a finite nonzero measure m BM on Γ\G T invariant by the geodesic flow (called the Bowen-Margulis measure). Furthermore, the G-invariant probability measure m X on X disintegrates with respect to the fiber bundle X = Γ\G → Γ\G T = Γ\G/A(O) defined by Γg → Γg * , over the probability measure Recall that for all x ∈ C, we have x − = g · ∞ = g * (−∞) and x + = g · 0 = g * (+∞) where g is the unique lift of x in C; furthermore g * (0) = g * = * is constant.…”

Section: Letmentioning

confidence: 99%

“…The commutativity of the diagram in Assertion ( 5) is inspired by, and closely related to, the commutativity of the diagram in [BP,Theo. 3.7].…”

Section: A Cross-section For the Orbits Of The Diagonal Groupmentioning

confidence: 99%

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On continued fraction expansions of quadratic irrationals in positive characteristic

Paulin

Shapira

2020

Groups Geom. Dyn.

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be the ring of polynomials over a finite field F q , let K = F q ((Y −1 )) be the field of formal Laurent series over F q , let f ∈ K be a quadratic irrational over F q (Y ) and let P ∈ R be an irreducible polynomial. We study the asymptotic properties of the degrees of the coefficients of the continued fraction expansion of quadratic irrationals such as P n f as n → +∞, proving, in sharp contrast with the case of quadratic irrationals in R over Q considered in [AS], that they have one such degree very large with respect to the other ones. We use arguments of [BP] giving a relationship with the discrete geodesic flow on the Bruhat-Tits building of (PGL 2 , K) and, with A the diagonal subgroup of PGL 2 ( K), the escape of mass phenomena of [KePS] for A-invariant probability measures on the compact A-orbits along Hecke rays in the moduli space PGL 2 (R)\ PGL 2 ( K). 1

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“…Then by [BP,Theo. 3.7] and Theorem 3 (2) and (5), the map Ξ : C → (R − F q ) Z defined by x → (a i ) i∈Z is a homeomorphism such that the following diagram commutes…”

Section: Letmentioning

confidence: 95%

Section: Letmentioning

confidence: 99%

Section: Letmentioning

confidence: 99%

“…The commutativity of the diagram in Assertion ( 5) is inspired by, and closely related to, the commutativity of the diagram in [BP,Theo. 3.7].…”

Section: A Cross-section For the Orbits Of The Diagonal Groupmentioning

confidence: 99%

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On continued fraction expansions of quadratic irrationals in positive characteristic

Paulin

Shapira

2020

Groups Geom. Dyn.

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“…Meanwhile, the discrete version of the geodesic flow is also explored by several authors ([ 2 , 3 , 4 ]). They considered the behavior of discrete geodesic flow system on and its application on Diophantine approximation over a positive characteristic field of formal series.…”

Section: Introductionmentioning

confidence: 99%

Measure Theoretic Entropy of Discrete Geodesic Flow on Nagao Lattice Quotient

Kwon

2020

Entropy

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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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Dynamique sur le rayon modulaire et fractions continues en caractéristique p

Cited by 10 publications

References 10 publications

On continued fraction expansions of quadratic irrationals in positive characteristic

On continued fraction expansions of quadratic irrationals in positive characteristic

Measure Theoretic Entropy of Discrete Geodesic Flow on Nagao Lattice Quotient

Equidistribution and Counting Under Equilibrium States in Negative Curvature and Trees

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