Geodesic flows, interval maps, and symbolic dynamics

Adler, Roy L.; Flatto, Leopold

doi:10.1090/s0273-0979-1991-16076-3

Cited by 142 publications

(212 citation statements)

References 33 publications

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“…The actions of 0 and λ on ∂D may be modelled by expanding Markov maps T 0 : ∂D → ∂D and T λ : ∂D → ∂D, respectively [1], [15]. These maps are conjugated by π λ : ∂D → ∂D, which is Hölder continuous.…”

Section: Proposition 24 We Have That Dmentioning

confidence: 99%

“…The result then follows by Proposition 2.4. is the Selberg zeta function which converges for Re(s) > 1 and has a simple zero at s = 1. Let us now consider the choice F : T 1 V → R given by F(v) = λg (1) The formulation of the · 2 WP in terms of the Selberg zeta function, combined with Ruelle's Grothendieck determinant approach to Z g (s) [13] leads, in principle, to a very efficient method for numerically computing the Weil-Petersson metric.…”

Section: Theorem 43 We Have Thatmentioning

confidence: 99%

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Weil–Petersson metrics, Manhattan curves and Hausdorff dimension

Pollicott

Sharp

2015

Math. Z.

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Section: Proposition 24 We Have That Dmentioning

confidence: 99%

Section: Theorem 43 We Have Thatmentioning

confidence: 99%

Weil–Petersson metrics, Manhattan curves and Hausdorff dimension

Pollicott

Sharp

2015

Math. Z.

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“…Barning [9], and later independently several others [7,11,12,15,16,18,21] (see also [20]), showed: In some of the papers where this was discussed, the theorem has been described as placing the PPTs (a, b, c) with a odd and b even on the nodes of an infinite rooted ternary tree, with the root representing the "basic" triple (3,4,5), and where each triple (a, b, c) has three children, representing the multiplication of the triple (considered as a column vector) by the three matrices M 1 , M 2 , M 3 . In this paper, we consider a slightly different outlook.…”

Section: Introductionmentioning

confidence: 99%

The dynamics of Pythagorean Triples

Romik

2008

Trans. Amer. Math. Soc.

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Abstract. We construct a piecewise onto 3-to-1 dynamical system on the positive quadrant of the unit circle, such that for rational points (which correspond to normalized Primitive Pythagorean Triples), the associated ternary expansion is finite and is equal to the address of the PPT on Barning's (1963) ternary tree of PPTs, while irrational points have infinite expansions. The dynamical system is conjugate to a modified Euclidean algorithm. The invariant measure is identified, and the system is shown to be conservative and ergodic. We also show, based on a result of Aaronson and Denker (1999), that the dynamical system can be obtained as a factor map of a cross section of the geodesic flow on a quotient space of the hyperbolic plane by the group Γ(2), a free subgroup of the modular group with two generators.

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“…Series, in [22], [23] and [24], and later, Adler and Flatto in [1], proved that T L (respectively T R ) is Markov with respect to a partition of The Ruelle transfer operator can be defined for any picewise C 2 Markov transformation (S 1 , T, {I k }) and any potential function A. Actually, we need a particular complex transfer operator given by the potential…”

Section: D Fs Can Be Extended To R As a 1 2 -Hölder Continuous Funmentioning

confidence: 99%

“…Let T L be the left Bowen-Series transformation that acts on the boundary S 1 = ∂D and is associated to a particular set of generators of Γ. The precise definition of T L has been given in [8], [22], [23], [24], and more geometrical descriptions have then been given in [1] and [18]. Specific examples of the Bowen-Series transformation have been studied in [17] and [4] for the modular surface and in [3] for a symmetric compact fundamental domain of genus two.…”

Section: Introductionmentioning

confidence: 99%

Eigenfunctions of the Laplacian and associated Ruelle operator

Lopes

Thieullen

2008

Nonlinearity

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Let Γ be a co-compact Fuchsian group of isometries on the Poincaré disk D and ∆ the corresponding hyperbolic Laplace operator. Any smooth eigenfunction f of ∆, equivariant by Γ with real eigenvalue λ = −s(1 − s), where s = 1 2 + it, admits an integral representation by a distribution D f,s (the Helgason distribution) which is equivariant by Γ and supported at infinity ∂D = S 1 . The geodesic flow on the compact surface D/Γ is conjugate to a suspension over a natural extension of a piecewise analytic map T : S 1 → S 1 , the so-called Bowen-Series transformation. Let L s be the complex Ruelle transfer operator associated to the jacobian −s ln |T ′ |. M. Pollicott showed that D f,s is an eigenfunction of the dual operator L * s for the eigenvalue 1. Here we show the existence of a (nonzero) piecewise real analytic eigenfunction ψ f,s of L s for the eigenvalue 1, given by an integral formulawhere J(ξ, η) is a {0, 1}-valued piecewise constant function whose definition depends upon the geometry of the Dirichlet fundamental domain representing the surface D/Γ. * Instituto de Matemática, UFRGS, Porto Alegre 91501-970, Brasil. Partially supported by CNPq, Fapergs, Instituto do Milenio, beneficiário de auxílio financeiro CAPES -Brasil.

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Geodesic flows, interval maps, and symbolic dynamics

Cited by 142 publications

References 33 publications

Weil–Petersson metrics, Manhattan curves and Hausdorff dimension

Weil–Petersson metrics, Manhattan curves and Hausdorff dimension

The dynamics of Pythagorean Triples

Eigenfunctions of the Laplacian and associated Ruelle operator

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