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)nH, 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)nH, 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)nH, 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)nH, 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)nH, 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