Geometric interpretations of continued fractions Natural extensions Conjugacy of the Rosen and Veech algorithms

Triangular billiards and generalized continued fractions

Pierre Arnoux

Orl´eans,26 mars 2008

Joint work with Thomas A. Schmidt

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Classical CF and their variants Natural extensions Rosen CF Conjugacy of the Rosen and Veech algorithms Veech CF Geometric Interpretations of continued fractions: Classical continued fractions

The classical expansion of real numbers in continued fraction admits several interpretations: Arithmetic: approximation by rational numbers pn I qn I Algebraic: matrices in GL(2, Z) I Geometric: approximation of a line in the plane by elements of 2 the lattice Z I Dynamic: geodesic flow on the modular surface SL(2, Z)\H, or equivalently Teichm¨uller flow for geometric structure for the torus.

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Classical CF and their variants Natural extensions Rosen CF Conjugacy of the Rosen and Veech algorithms Veech CF Geometric Interpretations of continued fractions: Classical continued fractions

The classical expansion of real numbers in continued fraction admits several interpretations: Arithmetic: approximation by rational numbers pn I qn I Algebraic: matrices in GL(2, Z) I Geometric: approximation of a line in the plane by elements of 2 the lattice Z I Dynamic: geodesic flow on the modular surface SL(2, Z)\H, or equivalently Teichm¨ullerflow for geometric structure for the torus.

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Classical CF and their variants Natural extensions Rosen CF Conjugacy of the Rosen and Veech algorithms Veech CF Geometric Interpretations of continued fractions: Classical continued fractions

The classical expansion of real numbers in continued fraction admits several interpretations: Arithmetic: approximation by rational numbers pn I qn I Algebraic: matrices in GL(2, Z) I Geometric: approximation of a line in the plane by elements of 2 the lattice Z I Dynamic: geodesic flow on the modular surface SL(2, Z)\H, or equivalently Teichm¨ullerflow for geometric structure for the torus.

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Classical CF and their variants Natural extensions Rosen CF Conjugacy of the Rosen and Veech algorithms Veech CF Geometric Interpretations of continued fractions: Classical continued fractions

The classical expansion of real numbers in continued fraction admits several interpretations: Arithmetic: approximation by rational numbers pn I qn I Algebraic: matrices in GL(2, Z) I Geometric: approximation of a line in the plane by elements of 2 the lattice Z I Dynamic: geodesic flow on the modular surface SL(2, Z)\H, or equivalently Teichm¨ullerflow for geometric structure for the torus.

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Classical CF and their variants Natural extensions Rosen CF Conjugacy of the Rosen and Veech algorithms Veech CF Geometric Interpretations of continued fractions: Classical continued fractions

The classical expansion of real numbers in continued fraction admits several interpretations: Arithmetic: approximation by rational numbers pn I qn I Algebraic: matrices in GL(2, Z) I Geometric: approximation of a line in the plane by elements of 2 the lattice Z I Dynamic: geodesic flow on the modular surface SL(2, Z)\H, or equivalently Teichm¨ullerflow for geometric structure for the torus.

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Classical CF and their variants Natural extensions Rosen CF Conjugacy of the Rosen and Veech algorithms Veech CF The Klein model

17.5

15

12.5

10

7.5

5

2.5

2 4 6 8 10 12

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Classical CF and their variants Natural extensions Rosen CF Conjugacy of the Rosen and Veech algorithms Veech CF

The classical domain for SL(2, Z)

Pierre Arnoux Triangular billiards and generalized continued fractions It is natural to look for such interpretations for generalized continued fractions.

Geometric interpretations of continued fractions Classical CF and their variants Natural extensions Rosen CF Conjugacy of the Rosen and Veech algorithms Veech CF Variants of the classical continued fraction

I Additive Continued Fraction

I Nearest Integer Continued Fraction

I Optimal Continued Fraction

I α-Continued Fraction (Japanese continued fractions) Occur naturally in various contexts. All linked to the group SL(2, Z), lead essentially to the same results (infinite number of common approximants, for example). They can be analyzed as first return maps of the modular flow on various sections.

Pierre Arnoux Triangular billiards and generalized continued fractions It is natural to look for such interpretations for generalized continued fractions.

Geometric interpretations of continued fractions Classical CF and their variants Natural extensions Rosen CF Conjugacy of the Rosen and Veech algorithms Veech CF Variants of the classical continued fraction

I Additive Continued Fraction

I Nearest Integer Continued Fraction

I Optimal Continued Fraction

I α-Continued Fraction (Japanese continued fractions) Occur naturally in various contexts. All linked to the group SL(2, Z), lead essentially to the same results (infinite number of common approximants, for example). They can be analyzed as first return maps of the modular flow on various sections.

Pierre Arnoux Triangular billiards and generalized continued fractions It is natural to look for such interpretations for generalized continued fractions.

Geometric interpretations of continued fractions Classical CF and their variants Natural extensions Rosen CF Conjugacy of the Rosen and Veech algorithms Veech CF Variants of the classical continued fraction

I Additive Continued Fraction

I Nearest Integer Continued Fraction

I Optimal Continued Fraction

I α-Continued Fraction (Japanese continued fractions) Occur naturally in various contexts. All linked to the group SL(2, Z), lead essentially to the same results (infinite number of common approximants, for example). They can be analyzed as first return maps of the modular flow on various sections.

Pierre Arnoux Triangular billiards and generalized continued fractions It is natural to look for such interpretations for generalized continued fractions.

Geometric interpretations of continued fractions Classical CF and their variants Natural extensions Rosen CF Conjugacy of the Rosen and Veech algorithms Veech CF Variants of the classical continued fraction

I Additive Continued Fraction

I Nearest Integer Continued Fraction

I Optimal Continued Fraction

I α-Continued Fraction (Japanese continued fractions) Occur naturally in various contexts. All linked to the group SL(2, Z), lead essentially to the same results (infinite number of common approximants, for example). They can be analyzed as first return maps of the modular flow on various sections.

Pierre Arnoux Triangular billiards and generalized continued fractions It is natural to look for such interpretations for generalized continued fractions.

Geometric interpretations of continued fractions Classical CF and their variants Natural extensions Rosen CF Conjugacy of the Rosen and Veech algorithms Veech CF Variants of the classical continued fraction

I Additive Continued Fraction

I Nearest Integer Continued Fraction

I Optimal Continued Fraction

I α-Continued Fraction (Japanese continued fractions) Occur naturally in various contexts. All linked to the group SL(2, Z), lead essentially to the same results (infinite number of common approximants, for example). They can be analyzed as first return maps of the modular flow on various sections.

Pierre Arnoux Triangular billiards and generalized continued fractions It is natural to look for such interpretations for generalized continued fractions.

Geometric interpretations of continued fractions Classical CF and their variants Natural extensions Rosen CF Conjugacy of the Rosen and Veech algorithms Veech CF Variants of the classical continued fraction

I Additive Continued Fraction

I Nearest Integer Continued Fraction

I Optimal Continued Fraction

I α-Continued Fraction (Japanese continued fractions) Occur naturally in various contexts. All linked to the group SL(2, Z), lead essentially to the same results (infinite number of common approximants, for example). They can be analyzed as first return maps of the modular flow on various sections.

Pierre Arnoux Triangular billiards and generalized continued fractions It is natural to look for such interpretations for generalized continued fractions.

Geometric interpretations of continued fractions Classical CF and their variants Natural extensions Rosen CF Conjugacy of the Rosen and Veech algorithms Veech CF Variants of the classical continued fraction

I Additive Continued Fraction

I Nearest Integer Continued Fraction

I Optimal Continued Fraction

I α-Continued Fraction (Japanese continued fractions) Occur naturally in various contexts. All linked to the group SL(2, Z), lead essentially to the same results (infinite number of common approximants, for example). They can be analyzed as first return maps of the modular flow on various sections.

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Classical CF and their variants Natural extensions Rosen CF Conjugacy of the Rosen and Veech algorithms Veech CF Variants of the classical continued fraction

I Additive Continued Fraction

I Nearest Integer Continued Fraction

I Optimal Continued Fraction

I α-Continued Fraction (Japanese continued fractions) Occur naturally in various contexts. All linked to the group SL(2, Z), lead essentially to the same results (infinite number of common approximants, for example). They can be analyzed as first return maps of the modular flow on various sections. It is natural to look for such interpretations for generalized continued fractions.

Pierre Arnoux Triangular billiards and generalized continued fractions π Specially interesting if λ = λq = 2 cos q Associated to the map: 1 λ λ λ λ x 7→ |x| − kx λ [− 2 , 2 [→ [− 2 , 2 [ 1 λ λ where kx is the unique integer k ∈ Z such that |x| − kλ ∈ [− 2 , 2 [

Geometric interpretations of continued fractions Classical CF and their variants Natural extensions Rosen CF Conjugacy of the Rosen and Veech algorithms Veech CF An arithmetic generalization: Rosen Continued fraction

In the nearest integer continued fraction, take for the partial quotients multiples of a fixed number λ instead of integers; i.e. write real numbers as:  x = n λ + 1 0 2 n1λ + 3 n2λ+ n3λ+...

with ni ∈ Z, i = ±1

Pierre Arnoux Triangular billiards and generalized continued fractions Associated to the map: 1 λ λ λ λ x 7→ |x| − kx λ [− 2 , 2 [→ [− 2 , 2 [ 1 λ λ where kx is the unique integer k ∈ Z such that |x| − kλ ∈ [− 2 , 2 [

Geometric interpretations of continued fractions Classical CF and their variants Natural extensions Rosen CF Conjugacy of the Rosen and Veech algorithms Veech CF An arithmetic generalization: Rosen Continued fraction

In the nearest integer continued fraction, take for the partial quotients multiples of a fixed number λ instead of integers; i.e. write real numbers as:  x = n λ + 1 0 2 n1λ + 3 n2λ+ n3λ+...

with ni ∈ Z, i = ±1 π Specially interesting if λ = λq = 2 cos q

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Classical CF and their variants Natural extensions Rosen CF Conjugacy of the Rosen and Veech algorithms Veech CF An arithmetic generalization: Rosen Continued fraction

In the nearest integer continued fraction, take for the partial quotients multiples of a fixed number λ instead of integers; i.e. write real numbers as:  x = n λ + 1 0 2 n1λ + 3 n2λ+ n3λ+...

with ni ∈ Z, i = ±1 π Specially interesting if λ = λq = 2 cos q Associated to the map: 1 λ λ λ λ x 7→ |x| − kx λ [− 2 , 2 [→ [− 2 , 2 [ 1 λ λ where kx is the unique integer k ∈ Z such that |x| − kλ ∈ [− 2 , 2 [

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Classical CF and their variants Natural extensions Rosen CF Conjugacy of the Rosen and Veech algorithms Veech CF

If λ = λq, SL(2, Z) is replaced by Hecke group Hq, generated by: 0 −1 1 λ  T = S = q 1 0 q 0 1 small problem with orientation, solved by symmetrization: −1 x 7→ − k λ x x written as x 7→ (S−k T ).x Problem: can we find a geometric interpretation of this generalized continued fraction, for example in terms of deformation of geometric structure (flow on moduli spaces)?

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Classical CF and their variants Natural extensions Rosen CF Conjugacy of the Rosen and Veech algorithms Veech CF

If λ = λq, SL(2, Z) is replaced by Hecke group Hq, generated by: 0 −1 1 λ  T = S = q 1 0 q 0 1 small problem with orientation, solved by symmetrization: −1 x 7→ − k λ x x written as x 7→ (S−k T ).x Problem: can we find a geometric interpretation of this generalized continued fraction, for example in terms of deformation of geometric structure (flow on moduli spaces)?

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Classical CF and their variants Natural extensions Rosen CF Conjugacy of the Rosen and Veech algorithms Veech CF

If λ = λq, SL(2, Z) is replaced by Hecke group Hq, generated by: 0 −1 1 λ  T = S = q 1 0 q 0 1 small problem with orientation, solved by symmetrization: −1 x 7→ − k λ x x written as x 7→ (S−k T ).x Problem: can we find a geometric interpretation of this generalized continued fraction, for example in terms of deformation of geometric structure (flow on moduli spaces)?

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Classical CF and their variants Natural extensions Rosen CF Conjugacy of the Rosen and Veech algorithms Veech CF

If λ = λq, SL(2, Z) is replaced by Hecke group Hq, generated by: 0 −1 1 λ  T = S = q 1 0 q 0 1 small problem with orientation, solved by symmetrization: −1 x 7→ − k λ x x written as x 7→ (S−k T ).x Problem: can we find a geometric interpretation of this generalized continued fraction, for example in terms of deformation of geometric structure (flow on moduli spaces)?

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Classical CF and their variants Natural extensions Rosen CF Conjugacy of the Rosen and Veech algorithms Veech CF

If λ = λq, SL(2, Z) is replaced by Hecke group Hq, generated by: 0 −1 1 λ  T = S = q 1 0 q 0 1 small problem with orientation, solved by symmetrization: −1 x 7→ − k λ x x written as x 7→ (S−k T ).x Problem: can we find a geometric interpretation of this generalized continued fraction, for example in terms of deformation of geometric structure (flow on moduli spaces)?

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Classical CF and their variants Natural extensions Rosen CF Conjugacy of the Rosen and Veech algorithms Veech CF

The domain for Hq

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Classical CF and their variants Natural extensions Rosen CF Conjugacy of the Rosen and Veech algorithms Veech CF Rosen symmetrized continued fraction

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Classical CF and their variants Natural extensions Rosen CF Conjugacy of the Rosen and Veech algorithms Veech CF Rational triangular billiard

π Billiard in a right triangle with one angle n The group generated by reflexions on the sides has order 2n By identifying sides, we obtain one or two regular polygon, depending on parity Opposite sides are identified

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Classical CF and their variants Natural extensions Rosen CF Conjugacy of the Rosen and Veech algorithms Veech CF Rational triangular billiard

π Billiard in a right triangle with one angle n The group generated by reflexions on the sides has order 2n By identifying sides, we obtain one or two regular polygon, depending on parity Opposite sides are identified

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Classical CF and their variants Natural extensions Rosen CF Conjugacy of the Rosen and Veech algorithms Veech CF Rational triangular billiard

π Billiard in a right triangle with one angle n The group generated by reflexions on the sides has order 2n By identifying sides, we obtain one or two regular polygon, depending on parity Opposite sides are identified

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Classical CF and their variants Natural extensions Rosen CF Conjugacy of the Rosen and Veech algorithms Veech CF Rational triangular billiard

π Billiard in a right triangle with one angle n The group generated by reflexions on the sides has order 2n By identifying sides, we obtain one or two regular polygon, depending on parity Opposite sides are identified

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Classical CF and their variants Natural extensions Rosen CF Conjugacy of the Rosen and Veech algorithms Veech CF π The billard with angle 8

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Classical CF and their variants Natural extensions Rosen CF Conjugacy of the Rosen and Veech algorithms Veech CF A geometric generalization: Veech Continued fraction

Regular polygon with q = 2n sides. By identification of opposite sides: Riemann surface with a canonical quadratic differential given by the natural flat structure out of the vertices of the polygon. This flat structure admits a non-trivial group of affine automorphisms, its Veech group Vq, which preserves the affine structure. π Fix µq = 2 cot q Both maps

2π 2π !   cos q − sin q 1 µq Rq = 2π 2π Pq = sin q − cos q 0 1

preserve the affine structure; they generate the Veech group Vq, subgroup of index 2 of a group Gq conjugate to Hq.

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Classical CF and their variants Natural extensions Rosen CF Conjugacy of the Rosen and Veech algorithms Veech CF A geometric generalization: Veech Continued fraction

Regular polygon with q = 2n sides. By identification of opposite sides: Riemann surface with a canonical quadratic differential given by the natural flat structure out of the vertices of the polygon. This flat structure admits a non-trivial group of affine automorphisms, its Veech group Vq, which preserves the affine structure. π Fix µq = 2 cot q Both maps

2π 2π !   cos q − sin q 1 µq Rq = 2π 2π Pq = sin q − cos q 0 1

preserve the affine structure; they generate the Veech group Vq, subgroup of index 2 of a group Gq conjugate to Hq.

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Classical CF and their variants Natural extensions Rosen CF Conjugacy of the Rosen and Veech algorithms Veech CF A geometric generalization: Veech Continued fraction

Regular polygon with q = 2n sides. By identification of opposite sides: Riemann surface with a canonical quadratic differential given by the natural flat structure out of the vertices of the polygon. This flat structure admits a non-trivial group of affine automorphisms, its Veech group Vq, which preserves the affine structure. π Fix µq = 2 cot q Both maps

2π 2π !   cos q − sin q 1 µq Rq = 2π 2π Pq = sin q − cos q 0 1

preserve the affine structure; they generate the Veech group Vq, subgroup of index 2 of a group Gq conjugate to Hq.

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Classical CF and their variants Natural extensions Rosen CF Conjugacy of the Rosen and Veech algorithms Veech CF A geometric generalization: Veech Continued fraction

Regular polygon with q = 2n sides. By identification of opposite sides: Riemann surface with a canonical quadratic differential given by the natural flat structure out of the vertices of the polygon. This flat structure admits a non-trivial group of affine automorphisms, its Veech group Vq, which preserves the affine structure. π Fix µq = 2 cot q Both maps

2π 2π !   cos q − sin q 1 µq Rq = 2π 2π Pq = sin q − cos q 0 1

preserve the affine structure; they generate the Veech group Vq, subgroup of index 2 of a group Gq conjugate to Hq.

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Classical CF and their variants Natural extensions Rosen CF Conjugacy of the Rosen and Veech algorithms Veech CF Renormalization of the octogon

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Classical CF and their variants Natural extensions Rosen CF Conjugacy of the Rosen and Veech algorithms Veech CF The domain of Veech group

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Classical CF and their variants Natural extensions Rosen CF Conjugacy of the Rosen and Veech algorithms Veech CF The Veech Continued fraction

The Veech group acts on the lines through the origin; one defines the Veech continued fraction, on the slope of these lines, in the following way: µq µq Line with slope in [− 2 , 2 ]. Act by Rq till the slope gets out of the interval. Then act by the parabolic element Pq till the slope comes back to the interval. One obtains in this way a generalized continued fraction, the additive Veech Algorithm We will prove that The Veech and Rosen continued fraction are essentially the same. More precisely: if we take a suitable conjugate the Rosen algorithm, then it has , for a.e. initial point, an infinite number of common iterates with Veech algorithm.

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Classical CF and their variants Natural extensions Rosen CF Conjugacy of the Rosen and Veech algorithms Veech CF The Veech Continued fraction

The Veech group acts on the lines through the origin; one defines the Veech continued fraction, on the slope of these lines, in the following way: µq µq Line with slope in [− 2 , 2 ]. Act by Rq till the slope gets out of the interval. Then act by the parabolic element Pq till the slope comes back to the interval. One obtains in this way a generalized continued fraction, the additive Veech Algorithm We will prove that The Veech and Rosen continued fraction are essentially the same. More precisely: if we take a suitable conjugate the Rosen algorithm, then it has , for a.e. initial point, an infinite number of common iterates with Veech algorithm.

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Classical CF and their variants Natural extensions Rosen CF Conjugacy of the Rosen and Veech algorithms Veech CF The Veech Continued fraction

The Veech group acts on the lines through the origin; one defines the Veech continued fraction, on the slope of these lines, in the following way: µq µq Line with slope in [− 2 , 2 ]. Act by Rq till the slope gets out of the interval. Then act by the parabolic element Pq till the slope comes back to the interval. One obtains in this way a generalized continued fraction, the additive Veech Algorithm We will prove that The Veech and Rosen continued fraction are essentially the same. More precisely: if we take a suitable conjugate the Rosen algorithm, then it has , for a.e. initial point, an infinite number of common iterates with Veech algorithm.

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Classical CF and their variants Natural extensions Rosen CF Conjugacy of the Rosen and Veech algorithms Veech CF The Veech Continued fraction

The Veech group acts on the lines through the origin; one defines the Veech continued fraction, on the slope of these lines, in the following way: µq µq Line with slope in [− 2 , 2 ]. Act by Rq till the slope gets out of the interval. Then act by the parabolic element Pq till the slope comes back to the interval. One obtains in this way a generalized continued fraction, the additive Veech Algorithm We will prove that The Veech and Rosen continued fraction are essentially the same. More precisely: if we take a suitable conjugate the Rosen algorithm, then it has , for a.e. initial point, an infinite number of common iterates with Veech algorithm.

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Classical CF and their variants Natural extensions Rosen CF Conjugacy of the Rosen and Veech algorithms Veech CF The Veech Continued fraction

The Veech group acts on the lines through the origin; one defines the Veech continued fraction, on the slope of these lines, in the following way: µq µq Line with slope in [− 2 , 2 ]. Act by Rq till the slope gets out of the interval. Then act by the parabolic element Pq till the slope comes back to the interval. One obtains in this way a generalized continued fraction, the additive Veech Algorithm We will prove that The Veech and Rosen continued fraction are essentially the same. More precisely: if we take a suitable conjugate the Rosen algorithm, then it has , for a.e. initial point, an infinite number of common iterates with Veech algorithm.

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Classical CF and their variants Natural extensions Rosen CF Conjugacy of the Rosen and Veech algorithms Veech CF Veech additive continued fraction

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Classical CF and their variants Natural extensions Rosen CF Conjugacy of the Rosen and Veech algorithms Veech CF Sketch of the proof

I Find a model for the natural extension in terms of sections for the geodesic flow in H\PSL(2, R) I Characterize first-return maps

I Prove that, after taking the square of the Rosen map and conjugating, these two maps are first return maps of the geodesic Teichm¨ullerflow on the Teichm¨ullerdisc of the regular polygon

I Consider their respective induction on the intersection of the definition domain: they are equal.

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Classical CF and their variants Natural extensions Rosen CF Conjugacy of the Rosen and Veech algorithms Veech CF Sketch of the proof

I Find a model for the natural extension in terms of sections for the geodesic flow in H\PSL(2, R) I Characterize first-return maps

I Prove that, after taking the square of the Rosen map and conjugating, these two maps are first return maps of the geodesic Teichm¨ullerflow on the Teichm¨ullerdisc of the regular polygon

I Consider their respective induction on the intersection of the definition domain: they are equal.

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Classical CF and their variants Natural extensions Rosen CF Conjugacy of the Rosen and Veech algorithms Veech CF Sketch of the proof

I Find a model for the natural extension in terms of sections for the geodesic flow in H\PSL(2, R) I Characterize first-return maps

I Prove that, after taking the square of the Rosen map and conjugating, these two maps are first return maps of the geodesic Teichm¨ullerflow on the Teichm¨ullerdisc of the regular polygon

I Consider their respective induction on the intersection of the definition domain: they are equal.

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Classical CF and their variants Natural extensions Rosen CF Conjugacy of the Rosen and Veech algorithms Veech CF Sketch of the proof

I Find a model for the natural extension in terms of sections for the geodesic flow in H\PSL(2, R) I Characterize first-return maps

I Prove that, after taking the square of the Rosen map and conjugating, these two maps are first return maps of the geodesic Teichm¨ullerflow on the Teichm¨ullerdisc of the regular polygon

I Consider their respective induction on the intersection of the definition domain: they are equal.

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Heuristics for natural extensions Natural extensions Parametrizations for SL(2, R) Conjugacy of the Rosen and Veech algorithms Numerical experiments Heuristics for models of natural extensions of piecewise homographies 2 Remark: everything is really done in PSL(2, R) = T1H ; for simplicity, we work in SL(2, R), allowing to replace M by −M. The Haar measure for SL(2, R) has a simple expression. 0 b In SL(2, ) the set of matrices has codimension 1, and R c d Haar measure 0. Out of this set, one can take coordinates (a, b, c) a b for the matrix . c d Lemma: In these coordinates, the Haar measure m, defined up to a constant, has expression: da db dc dm = a Proof is a trivial computation

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Heuristics for natural extensions Natural extensions Parametrizations for SL(2, R) Conjugacy of the Rosen and Veech algorithms Numerical experiments Heuristics for models of natural extensions of piecewise homographies 2 Remark: everything is really done in PSL(2, R) = T1H ; for simplicity, we work in SL(2, R), allowing to replace M by −M. The Haar measure for SL(2, R) has a simple expression. 0 b In SL(2, ) the set of matrices has codimension 1, and R c d Haar measure 0. Out of this set, one can take coordinates (a, b, c) a b for the matrix . c d Lemma: In these coordinates, the Haar measure m, defined up to a constant, has expression: da db dc dm = a Proof is a trivial computation

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Heuristics for natural extensions Natural extensions Parametrizations for SL(2, R) Conjugacy of the Rosen and Veech algorithms Numerical experiments Heuristics for models of natural extensions of piecewise homographies 2 Remark: everything is really done in PSL(2, R) = T1H ; for simplicity, we work in SL(2, R), allowing to replace M by −M. The Haar measure for SL(2, R) has a simple expression. 0 b In SL(2, ) the set of matrices has codimension 1, and R c d Haar measure 0. Out of this set, one can take coordinates (a, b, c) a b for the matrix . c d Lemma: In these coordinates, the Haar measure m, defined up to a constant, has expression: da db dc dm = a Proof is a trivial computation

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Heuristics for natural extensions Natural extensions Parametrizations for SL(2, R) Conjugacy of the Rosen and Veech algorithms Numerical experiments Heuristics for models of natural extensions of piecewise homographies 2 Remark: everything is really done in PSL(2, R) = T1H ; for simplicity, we work in SL(2, R), allowing to replace M by −M. The Haar measure for SL(2, R) has a simple expression. 0 b In SL(2, ) the set of matrices has codimension 1, and R c d Haar measure 0. Out of this set, one can take coordinates (a, b, c) a b for the matrix . c d Lemma: In these coordinates, the Haar measure m, defined up to a constant, has expression: da db dc dm = a Proof is a trivial computation

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Heuristics for natural extensions Natural extensions Parametrizations for SL(2, R) Conjugacy of the Rosen and Veech algorithms Numerical experiments The diagonal parametrization

et/2 0  Using the diagonal flow g = , one can write almost t 0 e−t/2 all element of PSL(2, R) as:

Xet/2 (XY − 1)e−t/2 et/2 Ye−t/2

In these coordinates, the Haar measure admits the simple expression: dm = dX dY dt

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Heuristics for natural extensions Natural extensions Parametrizations for SL(2, R) Conjugacy of the Rosen and Veech algorithms Numerical experiments The Hyperbolic plane parametrization

One can write almost any element as

t/2 −t/2 √Xe √Ye ! X −Y X −Y t/2 −t/2 √e √e X −Y X −Y In these coordinates, the Haar measure admits the expression: dX dY dt dm = (X − Y )2

This formula admits a nice interpretation as Liouville measure on the unit tangent of the hyperbolic plane.

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Heuristics for natural extensions Natural extensions Parametrizations for SL(2, R) Conjugacy of the Rosen and Veech algorithms Numerical experiments The arithmetic parametrization

1 Taking Z = − Y , one obtains an expression more common in arithmetics:

dX dZ dt dm = (1 + XZ)2

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Heuristics for natural extensions Natural extensions Parametrizations for SL(2, R) Conjugacy of the Rosen and Veech algorithms Numerical experiments The search for a natural extension: numerical experiments.

T : x 7→ Mx .x piecewise homography T on an interval. M is a piecewise constant map to PSL(2, R), the Mx generate a discrete subgroup Γ of PSL(2, R). We look for an explicit model of the natural extension T˜ . We look for this natural extension as a first-return map of the x (xy − 1) geodesic flow on Γ\ to the section , using the H 1 y diagonal parametrization and taking x as the first coordinate. A computation shows that the natural extension must be of the form:

ax + b  (x, y) 7→ , (cx + d)2y − c(cx + d) cx + d

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Heuristics for natural extensions Natural extensions Parametrizations for SL(2, R) Conjugacy of the Rosen and Veech algorithms Numerical experiments The search for a natural extension: numerical experiments.

T : x 7→ Mx .x piecewise homography T on an interval. M is a piecewise constant map to PSL(2, R), the Mx generate a discrete subgroup Γ of PSL(2, R). We look for an explicit model of the natural extension T˜ . We look for this natural extension as a first-return map of the x (xy − 1) geodesic flow on Γ\ to the section , using the H 1 y diagonal parametrization and taking x as the first coordinate. A computation shows that the natural extension must be of the form:

ax + b  (x, y) 7→ , (cx + d)2y − c(cx + d) cx + d

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Heuristics for natural extensions Natural extensions Parametrizations for SL(2, R) Conjugacy of the Rosen and Veech algorithms Numerical experiments The search for a natural extension: numerical experiments.

T : x 7→ Mx .x piecewise homography T on an interval. M is a piecewise constant map to PSL(2, R), the Mx generate a discrete subgroup Γ of PSL(2, R). We look for an explicit model of the natural extension T˜ . We look for this natural extension as a first-return map of the x (xy − 1) geodesic flow on Γ\ to the section , using the H 1 y diagonal parametrization and taking x as the first coordinate. A computation shows that the natural extension must be of the form:

ax + b  (x, y) 7→ , (cx + d)2y − c(cx + d) cx + d

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Heuristics for natural extensions Natural extensions Parametrizations for SL(2, R) Conjugacy of the Rosen and Veech algorithms Numerical experiments The search for a natural extension: numerical experiments.

T : x 7→ Mx .x piecewise homography T on an interval. M is a piecewise constant map to PSL(2, R), the Mx generate a discrete subgroup Γ of PSL(2, R). We look for an explicit model of the natural extension T˜ . We look for this natural extension as a first-return map of the x (xy − 1) geodesic flow on Γ\ to the section , using the H 1 y diagonal parametrization and taking x as the first coordinate. A computation shows that the natural extension must be of the form:

ax + b  (x, y) 7→ , (cx + d)2y − c(cx + d) cx + d

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Heuristics for natural extensions Natural extensions Parametrizations for SL(2, R) Conjugacy of the Rosen and Veech algorithms Numerical experiments The search for a natural extension: numerical experiments.

Problem: one does not know the domain of the natural extension. Solution: since we know the explicit form of the map, and since it is ergodic for Lebesgue measure, take a generic point and compute its orbit. It is then usually easy to find the domain of the natural extension. The simplest example is the classical continued fraction, where one recovers the Gauss measure. Some problems are encountered if there are indifferent fixed points: infinite measure. In this case, one needs to accelerate the algorithm near the fixed point.

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Heuristics for natural extensions Natural extensions Parametrizations for SL(2, R) Conjugacy of the Rosen and Veech algorithms Numerical experiments The search for a natural extension: numerical experiments.

Problem: one does not know the domain of the natural extension. Solution: since we know the explicit form of the map, and since it is ergodic for Lebesgue measure, take a generic point and compute its orbit. It is then usually easy to find the domain of the natural extension. The simplest example is the classical continued fraction, where one recovers the Gauss measure. Some problems are encountered if there are indifferent fixed points: infinite measure. In this case, one needs to accelerate the algorithm near the fixed point.

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Heuristics for natural extensions Natural extensions Parametrizations for SL(2, R) Conjugacy of the Rosen and Veech algorithms Numerical experiments The search for a natural extension: numerical experiments.

Problem: one does not know the domain of the natural extension. Solution: since we know the explicit form of the map, and since it is ergodic for Lebesgue measure, take a generic point and compute its orbit. It is then usually easy to find the domain of the natural extension. The simplest example is the classical continued fraction, where one recovers the Gauss measure. Some problems are encountered if there are indifferent fixed points: infinite measure. In this case, one needs to accelerate the algorithm near the fixed point.

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Heuristics for natural extensions Natural extensions Parametrizations for SL(2, R) Conjugacy of the Rosen and Veech algorithms Numerical experiments The search for a natural extension: numerical experiments.

Problem: one does not know the domain of the natural extension. Solution: since we know the explicit form of the map, and since it is ergodic for Lebesgue measure, take a generic point and compute its orbit. It is then usually easy to find the domain of the natural extension. The simplest example is the classical continued fraction, where one recovers the Gauss measure. Some problems are encountered if there are indifferent fixed points: infinite measure. In this case, one needs to accelerate the algorithm near the fixed point.

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Heuristics for natural extensions Natural extensions Parametrizations for SL(2, R) Conjugacy of the Rosen and Veech algorithms Numerical experiments Natural extension of Veech continued fraction. This algorithm has two indifferent fixed points at the boundary of the interval; one accelerates the algoritm on the corresponding continuity interval. define β = µ/2 · (3µ2 − 4)/(5µ2 + 4) and γ = µ/2 · (5µ2 + 4)/(3µ2 − 4). The domain ΩV of the model for the natural extension is bounded:  y = 1/(x + γ ) on [−µ/2, −β[;  Above by Under by y = 1/(x + µ/2) ) on [−β, µ/2[ .  y = 1/(x − µ/2 ) on [−µ/2, β[; 

y = 1/(x − γ ) on [β, µ/2[ ;  2 π  The area of Ωv is cv = 2 log 8 cos q .

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Heuristics for natural extensions Natural extensions Parametrizations for SL(2, R) Conjugacy of the Rosen and Veech algorithms Numerical experiments Natural extension of Veech continued fraction. This algorithm has two indifferent fixed points at the boundary of the interval; one accelerates the algoritm on the corresponding continuity interval. define β = µ/2 · (3µ2 − 4)/(5µ2 + 4) and γ = µ/2 · (5µ2 + 4)/(3µ2 − 4). The domain ΩV of the model for the natural extension is bounded:  y = 1/(x + γ ) on [−µ/2, −β[;  Above by Under by y = 1/(x + µ/2) ) on [−β, µ/2[ .  y = 1/(x − µ/2 ) on [−µ/2, β[; 

y = 1/(x − γ ) on [β, µ/2[ ;  2 π  The area of Ωv is cv = 2 log 8 cos q .

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Heuristics for natural extensions Natural extensions Parametrizations for SL(2, R) Conjugacy of the Rosen and Veech algorithms Numerical experiments Natural extension of Veech continued fraction. This algorithm has two indifferent fixed points at the boundary of the interval; one accelerates the algoritm on the corresponding continuity interval. define β = µ/2 · (3µ2 − 4)/(5µ2 + 4) and γ = µ/2 · (5µ2 + 4)/(3µ2 − 4). The domain ΩV of the model for the natural extension is bounded:  y = 1/(x + γ ) on [−µ/2, −β[;  Above by Under by y = 1/(x + µ/2) ) on [−β, µ/2[ .  y = 1/(x − µ/2 ) on [−µ/2, β[; 

y = 1/(x − γ ) on [β, µ/2[ ;  2 π  The area of Ωv is cv = 2 log 8 cos q .

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Heuristics for natural extensions Natural extensions Parametrizations for SL(2, R) Conjugacy of the Rosen and Veech algorithms Numerical experiments Veech multiplicative continued fraction

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Heuristics for natural extensions Natural extensions Parametrizations for SL(2, R) Conjugacy of the Rosen and Veech algorithms Numerical experiments Natural extension for Veech continued fraction

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Heuristics for natural extensions Natural extensions Parametrizations for SL(2, R) Conjugacy of the Rosen and Veech algorithms Numerical experiments Natural extension of Rosen continued fraction Using numerical experiments, one finds the domain of the natural extension; denoting m = M.1 its domain ΩR is bounded by:  y = 1 under [−2/µ, µ/2[ ;  x−m    y = 1 above [−µ/2, 2/µ[;  x+m  1 (j+1)π jπ y = j under [− tan q , − tan q [;  x−R 2 .m     1 jπ (j+1)π y = j above [tan q , tan q [ ,  x+R 2 .m The conjugate algorithm is of first return type. 1/2 It uses Rq ; one checks that its square only uses Rq and Pq, hence its associated natural extension is also of first return type.

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Heuristics for natural extensions Natural extensions Parametrizations for SL(2, R) Conjugacy of the Rosen and Veech algorithms Numerical experiments Natural extension for Rosen continued fraction

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Heuristics for natural extensions Natural extensions Parametrizations for SL(2, R) Conjugacy of the Rosen and Veech algorithms Numerical experiments Natural extension for Rosen continued fraction

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Functions of first-return type Natural extensions The conjugacy Conjugacy of the Rosen and Veech algorithms remarks Functions of first return type

Suppose that the map T is a piecewise isometry whose associated matrices generate a discret group Γ. Suppose that the natural extension T˜ is defined on a domain Ω, which can be considered as a surface in SL(2, R). We want to know whether T˜ is the first return map F to Ω of the geodesic flow on Γ. It is clear that one can write T˜ (P) = F nP (P). 1 Denote τ(x) = − 2 log(cx + d). One has: Z τ(x) dm ≥ πVol(Γ\PSL(2, R)) Ω

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Functions of first-return type Natural extensions The conjugacy Conjugacy of the Rosen and Veech algorithms remarks Functions of first return type

Theorem: T˜ is the first return map to the section if and only if Z τ(x) dm = πVol(Γ\PSL(2, R)) Ω One shows that the Veech algorithm is of first return type (not very surprising: it is defined in this way!) Theorem: if T is of first return type, T k is of first return type iff it generates a subgroup of index k of Γ.

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Functions of first-return type Natural extensions The conjugacy Conjugacy of the Rosen and Veech algorithms remarks Conjugacy of the Rosen and Veech algorithms

Rosen and Veech algorithm correspond to different groups, and are defined on different intervals. BUT groups Hq and Gq are conjugate by

1 1 cos π/q M = psin π/q 0 sin π/q

M sends interval [−λq/2, λq/2] to [0, µq]. Explicit conjugacy between Hq and Gq One obtains in this way a conjugate Rosen algorithm on [0, µq]; acting by Pq, one recovers an algorithm on [−µq/2, µq/2].

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Functions of first-return type Natural extensions The conjugacy Conjugacy of the Rosen and Veech algorithms remarks One has the identity:

−1 1/2 MTSqM = Rq . Using this identity, one computes the conjugate algorithm: k :[−µ/2, µ/2[→ [−µ/2, µ/2[ Which sends x to  1 1 µ − 2 2 R .x si 0 ≤ x < R . 2 ;     −k − 1 1 k µ 1 k µ P R 2 .x si R 2 P . − ≤ x < R 2 P . ;  2 2

 1 − 1 µ R 2 .x si 0 > x ≥ R 2 . − ;  2    1 1 µ 1 µ  l 2 − 2 −l − 2 −l P R .x si R P . − 2 ≤ x < R P . 2 .

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Functions of first-return type Natural extensions The conjugacy Conjugacy of the Rosen and Veech algorithms remarks Comparison of Rosen and Veech algorithms

The intersection Ω = ΩR ∩ ΩV is defined by: y = 1 over x ∈ [−µ/2, 2/µ[;  x+M.1    1 y = over x ∈ [ 2/µ, µ/2[ ;  x+µ/2

 1 y = x−µ/2 under x ∈ [−µ/2, −2/µ[;     1 y = x−M.1 under x ∈ [−2/µ, µ/2[ . From the preceding arguments, it is clear that the induced maps, on Ω, of Veech algorithm and the square of the conjugate Rosen algorithm are equal. By ergodicity, for almost all starting point, the two orbits have infinite intersection. Q.E.D.

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Functions of first-return type Natural extensions The conjugacy Conjugacy of the Rosen and Veech algorithms remarks Additional remarks

Which are the points with no common approximants? (essentially, closed trajectories that do not intersect the common section) It must be possible to give combinatorial rules: one of the viewpoints is that the two algorithms consist in taking a different set of generators for the group Vq. There is another algorithm, using interval exchanges on 4 intervals (for octogons, q = 8). This is similar to the link between continued fractions and rotations. The relation of this algorithm to the previous two is unclear. One should be in this setting able to use symbolic dynamics (like sturmian sequences for rotations). This should give some matrices√ in SL(4, Z), corresponding to the subgroup V8 of SL(2, Q( 2)).

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Functions of first-return type Natural extensions The conjugacy Conjugacy of the Rosen and Veech algorithms remarks Additional remarks

Which are the points with no common approximants? (essentially, closed trajectories that do not intersect the common section) It must be possible to give combinatorial rules: one of the viewpoints is that the two algorithms consist in taking a different set of generators for the group Vq. There is another algorithm, using interval exchanges on 4 intervals (for octogons, q = 8). This is similar to the link between continued fractions and rotations. The relation of this algorithm to the previous two is unclear. One should be in this setting able to use symbolic dynamics (like sturmian sequences for rotations). This should give some matrices√ in SL(4, Z), corresponding to the subgroup V8 of SL(2, Q( 2)).

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Functions of first-return type Natural extensions The conjugacy Conjugacy of the Rosen and Veech algorithms remarks Additional remarks

Which are the points with no common approximants? (essentially, closed trajectories that do not intersect the common section) It must be possible to give combinatorial rules: one of the viewpoints is that the two algorithms consist in taking a different set of generators for the group Vq. There is another algorithm, using interval exchanges on 4 intervals (for octogons, q = 8). This is similar to the link between continued fractions and rotations. The relation of this algorithm to the previous two is unclear. One should be in this setting able to use symbolic dynamics (like sturmian sequences for rotations). This should give some matrices√ in SL(4, Z), corresponding to the subgroup V8 of SL(2, Q( 2)).

Pierre Arnoux Triangular billiards and generalized continued fractions Geometric interpretations of continued fractions Functions of first-return type Natural extensions The conjugacy Conjugacy of the Rosen and Veech algorithms remarks Additional remarks

Which are the points with no common approximants? (essentially, closed trajectories that do not intersect the common section) It must be possible to give combinatorial rules: one of the viewpoints is that the two algorithms consist in taking a different set of generators for the group Vq. There is another algorithm, using interval exchanges on 4 intervals (for octogons, q = 8). This is similar to the link between continued fractions and rotations. The relation of this algorithm to the previous two is unclear. One should be in this setting able to use symbolic dynamics (like sturmian sequences for rotations). This should give some matrices√ in SL(4, Z), corresponding to the subgroup V8 of SL(2, Q( 2)).

Pierre Arnoux Triangular billiards and generalized continued fractions