Closed curves and geodesics with two self-intersections on the Punctured torus

Crisp, David J.; Dziadosz, Susan; Garity, Dennis J.; Insel, Thomas R.; Schmidt, Thomas A.; Wiles, Peter

doi:10.1007/bf01317313

Cited by 4 publications

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“…In their proof, [3] correctly show that each isometry class of the Type 6 geodesics (in their setting as geodesics on the punctured torus) can be represented by a purely periodic continued fraction, whose period has the form [5, 2, b 2 , b 3 , . .…”

Section: Example 1 High Type 4 Geodesics Consider the Wordmentioning

confidence: 93%

“…In fact, all closed Type 2 geodesics are in the same orbit under the automorphisms of Γ 3 \H (see [3,Proposition 7.5] for this). Thus, already in this setting, we have explicit examples underlying the fact (in terms used by [4]) that although the Markoff triples enumerate coset representatives of automorphisms mod-S GFG S G ulo isometries, they do not list all such cosets -only (exactly) enough to enumerate the isometry classes of simple closed geodesics (and therefore of the paired PSSI).…”

Section: Remarkmentioning

confidence: 99%

“…Earlier Example of Type 6. Following a suggestion of Moran, Crisp et al [3] classify the closed curves of two self-intersections on the punctured torus. As an application of this, they find the automorphism class of twice selfintersecting closed geodesics of heights between 3 and 6 on Γ \H .…”

Section: Example 1 High Type 4 Geodesics Consider the Wordmentioning

confidence: 99%

“…However, [3,Proposition 7.4] already shows that these geodesics have relatively large heights. Here, we must point out that their proof is flawed.…”

Section: Example 1 High Type 4 Geodesics Consider the Wordmentioning

confidence: 99%

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LOW HEIGHT GEODESICS ON $\Gamma^3 \backslash \mathcal{H}$: HEIGHT FORMULAS AND EXAMPLES

Schmidt

Sheingorn

2007

Int. J. Number Theory

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For Marvin Knopp, our teacher and loyal friend for 17 + 42 yearsThe Markoff spectrum of binary indefinite quadratic forms can be studied in terms of heights of geodesics on low-index covers of the modular surface. The lowest geodesics on Γ 3 \H are the simple closed geodesics; these are indexed up to isometry by Markoff triples of positive integers (x, y, z) with x 2 +y 2 +z 2 = 3xyz, and have heights p 9 − 4/z 2 . Geodesics considered by Crisp and Moran have heights p 9 + 4/z 2 ; they conjectured that these heights, which lie in the "mysterious region" between 3 and the Hall ray, are isolated in the Markoff Spectrum.In our previous work, we classified the low height-achieving non-simple geodesics of Γ 3 \H into seven types according to the topology of highest arcs. Here, we obtain explicit formulas for the heights of geodesics of the first three types; the conjecture holds for approximation by closed geodesics of any of these types. Explicit examples show that each of the remaining types is realized. Int. J. Number Theory 2007.03:475-501. Downloaded from www.worldscientific.com by UNIVERSITY OF CONNECTICUT on 02/03/15. For personal use only. Low Height Geodesics on Γ 3 \H: Height Formulas and Examples 477topology of this set: its isolated points, limit points and the like. We hence discuss certain limits in our final section. Remark on generality: Teichmüller space contextThe classification given in [11] applies mutatis mutandis to any hyperbolic punctured sphere with three elliptic points, see [12] or [9] for the necessary background for this generalization. We give explicit geodesics on Γ 3 \H; one can adjust our techniques to give corresponding geodesics on the general element of its Teichmüller space. However, out of concern for the length of these papers, we restrict ourselves to the case of number theoretic interest.

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Section: Example 1 High Type 4 Geodesics Consider the Wordmentioning

confidence: 93%

Section: Remarkmentioning

confidence: 99%

Section: Example 1 High Type 4 Geodesics Consider the Wordmentioning

confidence: 99%

“…However, [3,Proposition 7.4] already shows that these geodesics have relatively large heights. Here, we must point out that their proof is flawed.…”

Section: Example 1 High Type 4 Geodesics Consider the Wordmentioning

confidence: 99%

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LOW HEIGHT GEODESICS ON $\Gamma^3 \backslash \mathcal{H}$: HEIGHT FORMULAS AND EXAMPLES

Schmidt

Sheingorn

2007

Int. J. Number Theory

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“…Many natural generalizations and related topics are beyond the scope of this paper, for example the approximation of complex numbers [21] [70]. Do the methods presented here help to cover a larger part of the Markov and Lagrange spectra by considering more complicated geodesics [17] [18] [19]? Can one treat, say, ternary quadratic forms or binary cubic forms in a similar fashion?…”

Section: Introductionmentioning

confidence: 99%

The hyperbolic geometry of Markov’s theorem on Diophantine approximation and quadratic forms

Springborn

2018

Enseign. Math.

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Markov's theorem classifies the worst irrational numbers with respect to rational approximation and the indefinite binary quadratic forms whose values for integer arguments stay farthest away from zero. The main purpose of this paper is to present a new proof of Markov's theorem using hyperbolic geometry. The main ingredients are a dictionary to translate between hyperbolic geometry and algebra/number theory, and some very basic tools borrowed from modern geometric Teichmüller theory. Simple closed geodesics and ideal triangulations of the modular torus play an important role, and so does the problem: How far can a straight line crossing a triangle stay away from the vertices? Definite binary quadratic forms are briefly discussed in the last section. 11J06, 32G15arXiv:1702.05061v1 [math.GT] 16 Feb 2017It is fun and a recommended exercise to consider this question in elementary euclidean geometry. Here, we need to deal with ideal hyperbolic triangles, decorated with horocycles at the vertices, and "distance from the vertices" is to be understood as "signed distance from the horocycles" (Sec. 13).The subjects of this article, Diophantine approximation, quadratic forms, and the hyperbolic geometry of numbers, are connected with diverse areas of mathematics, ranging from from the phyllotaxis of plants [16] to the stability of the solar system [36], and from Gauss' Disquisitiones Arithmeticae to Mirzakhani's recent Fields Medal [51]. An adequate survey of this area, even if limited to the most important and most recent contributions, would be beyond the scope of this introduction. The books by Aigner [2] and Cassels [11] are excellent references for Markov's theorem, Bombieri [6] provides a concise proof, and more about the Markov and Lagrange spectra can be found in Malyshev's survey [46] and the book by Cusick and Flahive [20]. The following discussion focuses on a few historic sources and the most immediate context and is far from comprehensive.One can distinguish two approaches to a geometric treatment of continued fractions, Diophantine approximation, and quadratic forms. In both cases, number theory is connected to geometry by a common symmetry group, GL 2 ( ). The first approach, known as the geometry of numbers and connected with the name of Minkowski, deals with the geometry of the 2 -lattice. Klein interpreted continued fraction approximation, intuitively speaking, as "pulling a thread tight" around lattice points [40] [41]. This approach extends naturally to higher dimensions, leading to a multidimensional generalization of continued fractions that was championed by Arnold [3] [4]. Delone's comments on Markov's work [22] also belong in this category (see also [29]).In this article, we pursue the other approach involving Ford circles and the Farey tessellation of the hyperbolic plane (Fig. 6). This approach could be called the hyperbolic geometry of numbers. Before Ford's geometric proof [27] of Hurwitz's theorem [37] (Sec. 2), Speiser had apparently used the Ford circles to prove a weaker approximation theo...

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CLASSIFYING LOW HEIGHT GEODESICS ON $\Gamma^3 \backslash \mathcal{H}$

Schmidt

Sheingorn

2007

Int. J. Number Theory

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We show that low height-achieving non-simple geodesics on a low-index cover of the modular surface can be classified into seven types, according to the topology of highest arcs. The lowest geodesics of the signature (0;2,2,2,∞)-orbifold [Formula: see text] are the simple closed geodesics; these are indexed up to isometry by Markoff triples of positive integers (x, y, z) with x2 + y2 + z2 = 3xyz, and have heights [Formula: see text]. Geodesics considered by Crisp and Moran have heights [Formula: see text]; they conjectured that these heights, which lie in the "mysterious region" between 3 and the Hall ray, are isolated in the Markoff Spectrum. As a step in resolving this conjecture, we characterize the geometry on [Formula: see text] of geodesic arcs with heights strictly between 3 and 6. Of these, one type of geodesic arc cannot realize the height of any geodesic.

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Closed curves and geodesics with two self-intersections on the Punctured torus

Cited by 4 publications

References 10 publications

LOW HEIGHT GEODESICS ON $\Gamma^3 \backslash \mathcal{H}$: HEIGHT FORMULAS AND EXAMPLES

LOW HEIGHT GEODESICS ON $\Gamma^3 \backslash \mathcal{H}$: HEIGHT FORMULAS AND EXAMPLES

The hyperbolic geometry of Markov’s theorem on Diophantine approximation and quadratic forms

CLASSIFYING LOW HEIGHT GEODESICS ON $\Gamma^3 \backslash \mathcal{H}$

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