## Cutting Sequences for Geodesic Flow on the Modular Surface and Continued Fractions

Citations: | 10 - 0 self |

### BibTeX

@MISC{Grabiner_cuttingsequences,

author = {David J. Grabiner and Jeffrey and C. Lagarias},

title = {Cutting Sequences for Geodesic Flow on the Modular Surface and Continued Fractions},

year = {}

}

### OpenURL

### Abstract

Abstract. This paper describes the cutting sequences of geodesic flow on the modular surface H/PSL(2, Z) with respect to the standard fundamental domain F = {z = x+iy: −1 1 2 ≤ x ≤

### Citations

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Citation Context ... viewed as a generalization of the reduction theory of indefinite binary quadratic forms to arbitrary finitely-generated Fuchsian groups, see Katok [22]. More recently Bowen and Series [8] and Series =-=[43]-=-, [44] used “modified boundary expansions” to obtain a particularly simple symbolic expansion from which strong forms of ergodicity could be deduced. Adler and Flatto [2], [3] used “rectilinear map” c... |

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Citation Context ...inued fractions, and in 1935 G. Hedlund [16] used Artin’s coding to show that geodesic flow on this surface was ergodic. Some other symbolic encodings introduced were “boundary expansions” by Nielsen =-=[38]-=- and cutting sequences with respect to a fundamental domain of Γ by Koebe [25] and Morse [36]. Cutting sequence expansions can be viewed as a generalization of the reduction theory of indefinite binar... |

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Citation Context ...θ, written θ = [a0, a1, a2, · · ·], can be represented using 2 × 2 matrices. The use of such matrices to represent continued fractions appears in Frame [12] and Kolden [26], and is described in Stark =-=[48]-=-. Namely, one has (3.1) where pn qn [ a0 1 1 0 ] . . . [ an 1 1 0 ] = [ pn pn−1 = [a0, a1, . . . ,an] is the n-th convergent of θ. The ordinary continued fraction expansion for θ > 0 has a symbolic dy... |

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Citation Context ...pansions can be viewed as a generalization of the reduction theory of indefinite binary quadratic forms to arbitrary finitely-generated Fuchsian groups, see Katok [22]. More recently Bowen and Series =-=[8]-=- and Series [43], [44] used “modified boundary expansions” to obtain a particularly simple symbolic expansion from which strong forms of ergodicity could be deduced. Adler and Flatto [2], [3] used “re... |

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Citation Context ...d as a generalization of the reduction theory of indefinite binary quadratic forms to arbitrary finitely-generated Fuchsian groups, see Katok [22]. More recently Bowen and Series [8] and Series [43], =-=[44]-=- used “modified boundary expansions” to obtain a particularly simple symbolic expansion from which strong forms of ergodicity could be deduced. Adler and Flatto [2], [3] used “rectilinear map” codings... |

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Citation Context ...nformation about this Riemann surface. Symbolic codings of geodesics were introduced by Hadamard [14] in 1898 as a way of understanding the complicated motions on geodesics on such surfaces. E. Artin =-=[5]-=- showed in 1924 that there exist dense geodesics on the modular surface H/PSL(2, Z), using symbolic encodings with continued fractions, and in 1935 G. Hedlund [16] used Artin’s coding to show that geo... |

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Citation Context ...on given in Theorem 4.1. Theorem 5.1. Given θ = [a0; a1, a2, a3, · · ·] with −1 1 < θ < 2 2 and suppose an+1 = 1. Set αn = [0, an, an−1, . . . ,a1] = qn−1 , and βn = [an+1, an+2, · · ·], so that αn ∈ =-=[0, 1]-=- qn24 DAVID J. GRABINER AND JEFFREY C. LAGARIAS and βn ∈ [1, 2]. In terms of the linear fractional transformation [ ] 1 2 z + 2 N(z) := (z) = 2 1 2z + 1 , (5.1) the Minkowski geodesic continued fract... |

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Citation Context ... {z ∈ C : Im(z) > 0}. We study in particular the cutting sequences of vertical geodesics, {θ + it : t > 0} for θ ∈ R, which are related to a continued fraction expansion introduced in 1850 by Hermite =-=[18]-=- in terms of quadratic forms, and studied by Humbert [20, 21]. Our motivation for studying these cutting sequences arose from the problem of determining the legal continued fraction expansions for the... |

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Citation Context ...tudying these cutting sequences arose from the problem of determining the legal continued fraction expansions for the one-dimensional case of a multidimensional continued fraction studied in Lagarias =-=[28]-=-. This expansion, called the Minkowski geodesic continued fraction expansion (MGCF expansion), is based on following a parametrized family of lattice bases in GL(d + 1, Z)\GL(d + 1, R) as[ the paramet... |

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Citation Context ...al cutting sequences (Theorem 7.2). The totality of all possible convex polygons P that are fundamental domains of some group Γ has an explicit characterization. A general treatment appears in Maskit =-=[32]-=-, which covers non-convex fundamental domains, and also covers groups of isometries of H, which may include orientation-reversing isometries. Maskit [32, section 2] gives a sufficient condition for P ... |

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Citation Context ...d Series [8] and Series [43], [44] used “modified boundary expansions” to obtain a particularly simple symbolic expansion from which strong forms of ergodicity could be deduced. Adler and Flatto [2], =-=[3]-=- used “rectilinear map” codings of geodesic flow on H/Γ to explain why certain specific maps of the interval, such as the continued fraction map and backwards continued fraction map, have invariant me... |

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Citation Context ...quence encoding of a geodesic. The pictured geodesic has cutting sequence . . . , g0, g1, g2, . . ., starting with g1 from P. . g3 . g −1 3 . . . . ❘ geodesic enters. (This is the convention of Katok =-=[24]-=-; and opposite to that of Adler and Flatto [3, p. 243], who use the symbol gi on the exit edge.) This convention yields: Lemma 2.1. The cutting sequence (g1, . . . , gj) follows a geodesic from P to t... |

11 |
Cross section map for geodesic flow on the modular surface
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Citation Context ...en and Series [8] and Series [43], [44] used “modified boundary expansions” to obtain a particularly simple symbolic expansion from which strong forms of ergodicity could be deduced. Adler and Flatto =-=[2]-=-, [3] used “rectilinear map” codings of geodesic flow on H/Γ to explain why certain specific maps of the interval, such as the continued fraction map and backwards continued fraction map, have invaria... |

10 |
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Citation Context ...ith L. 3.4. Finite Automata. By a finite automaton we mean a deterministic finite-state automaton, as defined in Hopcroft and Ullman [19]. We consider finite automata used as transducers, as in Raney =-=[40]-=-, in which each transition edge of the automaton specifies the printing of a specific finite string of output letters, possibly empty. The automaton starts in an initial state, and the input sequence ... |

9 |
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Citation Context ... geodesics on such surfaces. E. Artin [5] showed in 1924 that there exist dense geodesics on the modular surface H/PSL(2, Z), using symbolic encodings with continued fractions, and in 1935 G. Hedlund =-=[16]-=- used Artin’s coding to show that geodesic flow on this surface was ergodic. Some other symbolic encodings introduced were “boundary expansions” by Nielsen [38] and cutting sequences with respect to a... |

9 |
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Citation Context ...tinuously on H, and then specialize to standard fundamental domain F of the modular group PSL(2, Z). General references for geodesic flow and for cutting sequences are Adler and Flatto [3] and Series =-=[46]-=-. 2.1. Cutting Sequence Shifts. Let H denote the hyperbolic plane, represented as the upper half-plane H = {z = x + iy : y > 0} with hyperbolic line element ds 2 = 1 y 2(dx 2 + dy 2 ) and volume dxdy ... |

7 |
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Citation Context ...tion (OCF) expansion for a real θ, written θ = [a0, a1, a2, · · ·], can be represented using 2 × 2 matrices. The use of such matrices to represent continued fractions appears in Frame [12] and Kolden =-=[26]-=-, and is described in Stark [48]. Namely, one has (3.1) where pn qn [ a0 1 1 0 ] . . . [ an 1 1 0 ] = [ pn pn−1 = [a0, a1, . . . ,an] is the n-th convergent of θ. The ordinary continued fraction expan... |

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Citation Context ... using a finite automaton; see section 3.4. 3.2. Minkowski Geodesic Continued Fraction Expansions. The Minkowski geodesic multidimensional continued fraction (MGCF) expansion is described in Lagarias =-=[27]-=-. We consider here the one-dimensional case. If − 1 1 < θ < it follows a 2 2 parametrized series of lattice bases [ ] 1 0 (3.8) Bt(θ) = −θ t for the parametrized family of lattices Λt = Z[(1, 0), (−θ,... |

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Citation Context ...] and Morse [36]. Cutting sequence expansions can be viewed as a generalization of the reduction theory of indefinite binary quadratic forms to arbitrary finitely-generated Fuchsian groups, see Katok =-=[22]-=-. More recently Bowen and Series [8] and Series [43], [44] used “modified boundary expansions” to obtain a particularly simple symbolic expansion from which strong forms of ergodicity could be deduced... |

4 |
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Citation Context ... sequence encodes the steps of subtraction needed in the division process to encode the ordinary continued fraction of θ; it also essentially gives all the intermediate convergents to θ, cf. Richards =-=[41]-=-, Theorem 2.1. For θ > 0 the set of allowable symbol sequences for the additive expansion is the full one-sided shift on two letters {R, D}. The association to the Farey tree is described in Lagarias ... |

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2 |
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Citation Context ...or they are identical for all Riemann surfaces of the same genus. (See Theorem 8.4 of [3].) Finally we remark that other encodings of geodesic flow on the modular surface H/PSL(2, Z) appear in Arnoux =-=[4]-=- and Lagarias and Pollington [29]. Cutting sequence encodings are of special interest because they are related to good Diophantine approximations and also apparently encode more information about geod... |

2 |
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Citation Context ...sequences of vertical geodesics, {θ + it : t > 0} for θ ∈ R, which are related to a continued fraction expansion introduced in 1850 by Hermite [18] in terms of quadratic forms, and studied by Humbert =-=[20, 21]-=-. Our motivation for studying these cutting sequences arose from the problem of determining the legal continued fraction expansions for the one-dimensional case of a multidimensional continued fractio... |

2 |
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Citation Context ...emann surfaces of the same genus. (See Theorem 8.4 of [3].) Finally we remark that other encodings of geodesic flow on the modular surface H/PSL(2, Z) appear in Arnoux [4] and Lagarias and Pollington =-=[29]-=-. Cutting sequence encodings are of special interest because they are related to good Diophantine approximations and also apparently encode more information about geodesic flow than some of these othe... |

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1 |
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Citation Context ...y continued fraction (OCF) expansion for a real θ, written θ = [a0, a1, a2, · · ·], can be represented using 2 × 2 matrices. The use of such matrices to represent continued fractions appears in Frame =-=[12]-=- and Kolden [26], and is described in Stark [48]. Namely, one has (3.1) where pn qn [ a0 1 1 0 ] . . . [ an 1 1 0 ] = [ pn pn−1 = [a0, a1, . . . ,an] is the n-th convergent of θ. The ordinary continue... |

1 |
1898) Les surfaces á corbures opposseés et leurs lignes geódésiques
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Citation Context ... minus a finite number of punctures, and geodesic flow on H/Γ contains both topological and conformal information about this Riemann surface. Symbolic codings of geodesics were introduced by Hadamard =-=[14]-=- in 1898 as a way of understanding the complicated motions on geodesics on such surfaces. E. Artin [5] showed in 1924 that there exist dense geodesics on the modular surface H/PSL(2, Z), using symboli... |

1 |
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(Show Context)
Citation Context ...sequences of vertical geodesics, {θ + it : t > 0} for θ ∈ R, which are related to a continued fraction expansion introduced in 1850 by Hermite [18] in terms of quadratic forms, and studied by Humbert =-=[20, 21]-=-. Our motivation for studying these cutting sequences arose from the problem of determining the legal continued fraction expansions for the one-dimensional case of a multidimensional continued fractio... |

1 |
Riemannsche Manningfaltigkeiten und nichteucklidische Raumfornen (Vierte Mitteilungen: Verlauf Geodatischer
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Citation Context ...geodesic flow on this surface was ergodic. Some other symbolic encodings introduced were “boundary expansions” by Nielsen [38] and cutting sequences with respect to a fundamental domain of Γ by Koebe =-=[25]-=- and Morse [36]. Cutting sequence expansions can be viewed as a generalization of the reduction theory of indefinite binary quadratic forms to arbitrary finitely-generated Fuchsian groups, see Katok [... |