July  2020, 40(7): 4341-4378. doi: 10.3934/dcds.2020183

Billiards on pythagorean triples and their Minkowski functions

Department of Mathematics, Computer Science and Physics, University of Udine, via delle Scienze 206, 33100 Udine, Italy

Received  July 2019 Revised  January 2020 Published  April 2020

Fund Project: The author is partially supported by the research project SiDiA of the University of Udine

It has long been known that the set of primitive pythagorean triples can be enumerated by descending certain ternary trees. We unify these treatments by considering hyperbolic billiard tables in the Poincaré disk model. Our tables have $ m\ge3 $ ideal vertices, and are subject to the restriction that reflections in the table walls are induced by matrices in the triangle group $ {\rm{PSU}}^\pm_{1,1} \mathbb{Z}[i] $. The resulting billiard map $ \widetilde B $ acts on the de Sitter space $ x_1^2+x_2^2-x_3^2 = 1 $, and has a natural factor $ B $ on the unit circle, the pythagorean triples appearing as the $ B $-preimages of fixed points. We compute the invariant densities of these maps, and prove the Lagrange and Galois theorems: A complex number of unit modulus has a preperiodic (purely periodic) $ B $-orbit precisely when it is quadratic (and isolated from its conjugate by a billiard wall) over $ \mathbb{Q}(i) $.

Each $ B $ as above is a $ (m-1) $-to-$ 1 $ orientation-reversing covering map of the circle, a property shared by the group character $ T(z) = z^{-(m-1)} $. We prove that there exists a homeomorphism $ \Phi $, unique up to postcomposition with elements in a dihedral group, that conjugates $ B $ with $ T $; in particular $ \Phi $ -whose prototype is the classical Minkowski question mark function- establishes a bijection between the set of points of degree $ \le2 $ over $ \mathbb{Q}(i) $ and the torsion subgroup of the circle. We provide an explicit formula for $ \Phi $, and prove that $ \Phi $ is singular and Hölder continuous with exponent $ \log(m-1) $ divided by the maximal periodic mean free path in the associated billiard table.

Citation: Giovanni Panti. Billiards on pythagorean triples and their Minkowski functions. Discrete & Continuous Dynamical Systems - A, 2020, 40 (7) : 4341-4378. doi: 10.3934/dcds.2020183
References:
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J. Aaronson, An Introduction to Infinite Ergodic Theory, Vol. 50, Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1997. doi: 10.1090/surv/050.  Google Scholar

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J. Aaronson and M. Denker, The Poincaré series of $\mathbb C\setminus\mathbb Z$, Ergodic Theory Dynam. Systems, 19 (1999), 1-20.  doi: 10.1017/S0143385799126592.  Google Scholar

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R. C. Alperin, The modular tree of Pythagoras, Amer. Math. Monthly, 112 (2005), 807-816.  doi: 10.1080/00029890.2005.11920254.  Google Scholar

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F. J. M. Barning, On Pythagorean and quasi-Pythagorean triangles and a generation process with the help of unimodular matrices, Math. Centrum Amsterdam Afd. Zuivere Wisk., 1963 (1963), 37 pp.  Google Scholar

[5]

M. A. Berger and Y. Wang, Bounded semigroups of matrices, Linear Algebra Appl., 166 (1992), 21-27.  doi: 10.1016/0024-3795(92)90267-E.  Google Scholar

[6]

B. Berggren, Pytagoreiska trianglar, Tidskrift för elementär matematik, fysik och kemi, 17 (1934), 129–139. Google Scholar

[7]

F. P. Boca and C. Linden, On Minkowski type question mark functions associated with even or odd continued fractions, Monatsh. Math., 187 (2018), 35-57.  doi: 10.1007/s00605-018-1205-8.  Google Scholar

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A. I. Borevich and I. R. Shafarevich, Number Theory, Vol. 20, Pure and Applied Mathematics, Academic Press, New York-London, 1966.  Google Scholar

[9]

T. Bousch and J. Mairesse, Asymptotic height optimization for topical IFS, Tetris heaps, and the finiteness conjecture, J. Amer. Math. Soc., 15 (2002), 77-111.  doi: 10.1090/S0894-0347-01-00378-2.  Google Scholar

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J. W. Cannon, W. J. Floyd, R. Kenyon and W. R. Parry, Hyperbolic geometry, in Flavors of Geometry, Vol. 31, Math. Sci. Res. Inst. Publ., Cambridge Univ. Press, Cambridge, 1997.  Google Scholar

[11]

D. Cass and P. J. Arpaia, Matrix generation of Pythagorean $n$-tuples, Proc. Amer. Math. Soc., 109 (1990), 1-7.  doi: 10.2307/2048355.  Google Scholar

[12]

S. CastleN. Peyerimhoff and K. F. Siburg, Billiards in ideal hyperbolic polygons, Discrete Contin. Dyn. Syst., 29 (2011), 893-908.  doi: 10.3934/dcds.2011.29.893.  Google Scholar

[13]

B. Cha and D. H. Kim, Lagrange spectrum of Romik's dynamical system, preprint, arXiv: 1903.02882. Google Scholar

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B. Cha and D. H. Kim, Number theoretical properties of Romik's dynamical system, Bull. Korean Math. Soc., 57 (2020), 251-274.  doi: 10.4134/BKMS.b190163.  Google Scholar

[15]

B. ChaE. Nguyen and B. Tauber, Quadratic forms and their Berggren trees, J. Number Theory, 185 (2018), 218-256.  doi: 10.1016/j.jnt.2017.09.003.  Google Scholar

[16]

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[17]

K. T. Conrad, Pythagorean descent, Semantic Scholar, 2007. Google Scholar

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I. P. Cornfeld, S. V. Fomin and Y. G. Sinaĭ, Ergodic Theory, Vol. 245, Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4615-6927-5.  Google Scholar

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A. Denjoy, Sur une fonction réelle de Minkowski, J. Math. Pures Appl., 17 (1938), 105-155.   Google Scholar

[20]

E. J. Eckert, The group of primitive Pythagorean triangles, Math. Mag., 57 (1984), 22-27.  doi: 10.1080/0025570X.1984.11977070.  Google Scholar

[21]

L. Elsner, The generalized spectral-radius theorem: An analytic-geometric proof, Linear Algebra Appl., 220 (1995), 151-159.  doi: 10.1016/0024-3795(93)00320-Y.  Google Scholar

[22]

D. Fried, Symbolic dynamics for triangle groups, Invent. Math., 125 (1996), 487-521.  doi: 10.1007/s002220050084.  Google Scholar

[23]

N. Guglielmi and M. Zennaro, Stability of linear problems: Joint spectral radius of sets of matrices, in Current Challenges in Stability Issues for Numerical Differential Equations, Vol. 2082, Lecture Notes in Math., Springer, Cham, 2014. doi: 10.1007/978-3-319-01300-8_5.  Google Scholar

[24]

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th edition, The Clarendon Press, Oxford University Press, New York, 1979.  Google Scholar

[25]

S. Isola, From infinite ergodic theory to number theory (and possibly back), Chaos Solitons Fractals, 44 (2011), 467-479.  doi: 10.1016/j.chaos.2011.01.015.  Google Scholar

[26]

O. Jenkinson and M. Pollicott, Joint spectral radius, Sturmian measures and the finiteness conjecture, Ergodic Theory Dynam. Systems, 38 (2018), 3062-3100.  doi: 10.1017/etds.2017.18.  Google Scholar

[27]

T. Jordan and T. Sahlsten, Fourier transforms of Gibbs measures for the Gauss map, Math. Ann., 364 (2016), 983-1023.  doi: 10.1007/s00208-015-1241-9.  Google Scholar

[28]

S. Katok, Fuchsian groups, geodesic flows on surfaces of constant negative curvature and symbolic coding of geodesics, in Homogeneous Flows, Moduli Spaces and Arithmetic, Vol. 10, Clay Math. Proc., Amer. Math. Soc., 2010.  Google Scholar

[29]

M. Kesseböhmer and B. O. Stratmann, A multifractal analysis for Stern-Brocot intervals, continued fractions and Diophantine growth rates, J. Reine Angew. Math., 605 (2007), 133-163.  doi: 10.1515/CRELLE.2007.029.  Google Scholar

[30]

R. Kołodziej, An infinite smooth invariant measure for some transformation of a circle, Bull. Acad. Polon. Sci. Sér. Sci. Math., 29 (1981), 549-551.   Google Scholar

[31]

J. C. Lagarias and Y. Wang, The finiteness conjecture for the generalized spectral radius of a set of matrices, Linear Algebra Appl., 214 (1995), 17-42.  doi: 10.1016/0024-3795(93)00052-2.  Google Scholar

[32]

C. Maclachlan and A. W. Reid, The Arithmetic of Hyperbolic 3-manifolds, Vol. 219, Graduate Texts in Mathematics, Springer-Verlag, New York, 2003. doi: 10.1007/978-1-4757-6720-9.  Google Scholar

[33]

A. Miller, Trees of integral triangles with given rectangular defect, Discrete Math., 313 (2013), 50-66.  doi: 10.1016/j.disc.2012.09.013.  Google Scholar

[34]

M. Misiurewicz, The result of Rafał Kołodziej, in Ergodic Theory (Sem., Les Plans-sur-Bex, 1980), Vol. 29, Monograph. Enseign. Math., Univ. Genève, Geneva, 1981.  Google Scholar

[35]

J. Morita, A transformation group of the Pythagorean numbers, Tsukuba J. Math., 10 (1986), 151-153.  doi: 10.21099/tkbjm/1496160398.  Google Scholar

[36]

U. Moschella, The de Sitter and anti-de Sitter sightseeing tour, in Einstein, 1905–2005, Vol. 47, Prog. Math. Phys., Birkhäuser, Basel, 2006. doi: 10.1007/3-7643-7436-5_4.  Google Scholar

[37]

K. Nomizu, The Lorentz-Poincaré metric on the upper half-space and its extension, Hokkaido Math. J., 11 (1982), 253-261.  doi: 10.14492/hokmj/1381757803.  Google Scholar

[38]

G. Panti, A general Lagrange theorem, Amer. Math. Monthly, 116 (2009), 70-74.  doi: 10.1080/00029890.2009.11920912.  Google Scholar

[39]

G. Panti, Slow continued fractions, transducers, and the Serret theorem, J. Number Theory, 185 (2018), 121-143.  doi: 10.1016/j.jnt.2017.08.034.  Google Scholar

[40]

A. M. Rockett and P. Szüsz, Continued Fractions, World Scientific Publishing Co., Inc., River Edge, NJ, 1992. doi: 10.1142/1725.  Google Scholar

[41]

V. A. Rohlin, Exact endomorphisms of a Lebesgue space, Izv. Akad. Nauk SSSR Ser. Mat., 25 (1961), 499-530.   Google Scholar

[42]

D. Romik, The dynamics of Pythagorean triples, Trans. Amer. Math. Soc., 360 (2008), 6045-6064.  doi: 10.1090/S0002-9947-08-04467-X.  Google Scholar

[43]

R. Salem, On some singular monotonic functions which are strictly increasing, Trans. Amer. Math. Soc., 53 (1943), 427-439.  doi: 10.1090/S0002-9947-1943-0007929-6.  Google Scholar

[44]

D. Singerman, Finitely maximal Fuchsian groups, J. London Math. Soc., 6 (1972), 29-38.  doi: 10.1112/jlms/s2-6.1.29.  Google Scholar

[45]

J. Smillie and C. Ulcigrai, Geodesic flow on the Teichmüller disk of the regular octagon, cutting sequences and octagon continued fractions maps, in Dynamical Numbers–Interplay Between Dynamical Systems and Number Theory, Vol. 532, Contempt. Math, Amer. Math. Soc., Providence, RI, 2010. doi: 10.1090/conm/532/10482.  Google Scholar

[46]

K. Takeuchi, Arithmetic triangle groups, J. Math. Soc. Japan, 29 (1977), 91-106.  doi: 10.2969/jmsj/02910091.  Google Scholar

[47]

M. Thaler, Transformations on $[0, \, 1]$ with infinite invariant measures, Israel J. Math., 46 (1983), 67-96.  doi: 10.1007/BF02760623.  Google Scholar

show all references

References:
[1]

J. Aaronson, An Introduction to Infinite Ergodic Theory, Vol. 50, Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1997. doi: 10.1090/surv/050.  Google Scholar

[2]

J. Aaronson and M. Denker, The Poincaré series of $\mathbb C\setminus\mathbb Z$, Ergodic Theory Dynam. Systems, 19 (1999), 1-20.  doi: 10.1017/S0143385799126592.  Google Scholar

[3]

R. C. Alperin, The modular tree of Pythagoras, Amer. Math. Monthly, 112 (2005), 807-816.  doi: 10.1080/00029890.2005.11920254.  Google Scholar

[4]

F. J. M. Barning, On Pythagorean and quasi-Pythagorean triangles and a generation process with the help of unimodular matrices, Math. Centrum Amsterdam Afd. Zuivere Wisk., 1963 (1963), 37 pp.  Google Scholar

[5]

M. A. Berger and Y. Wang, Bounded semigroups of matrices, Linear Algebra Appl., 166 (1992), 21-27.  doi: 10.1016/0024-3795(92)90267-E.  Google Scholar

[6]

B. Berggren, Pytagoreiska trianglar, Tidskrift för elementär matematik, fysik och kemi, 17 (1934), 129–139. Google Scholar

[7]

F. P. Boca and C. Linden, On Minkowski type question mark functions associated with even or odd continued fractions, Monatsh. Math., 187 (2018), 35-57.  doi: 10.1007/s00605-018-1205-8.  Google Scholar

[8]

A. I. Borevich and I. R. Shafarevich, Number Theory, Vol. 20, Pure and Applied Mathematics, Academic Press, New York-London, 1966.  Google Scholar

[9]

T. Bousch and J. Mairesse, Asymptotic height optimization for topical IFS, Tetris heaps, and the finiteness conjecture, J. Amer. Math. Soc., 15 (2002), 77-111.  doi: 10.1090/S0894-0347-01-00378-2.  Google Scholar

[10]

J. W. Cannon, W. J. Floyd, R. Kenyon and W. R. Parry, Hyperbolic geometry, in Flavors of Geometry, Vol. 31, Math. Sci. Res. Inst. Publ., Cambridge Univ. Press, Cambridge, 1997.  Google Scholar

[11]

D. Cass and P. J. Arpaia, Matrix generation of Pythagorean $n$-tuples, Proc. Amer. Math. Soc., 109 (1990), 1-7.  doi: 10.2307/2048355.  Google Scholar

[12]

S. CastleN. Peyerimhoff and K. F. Siburg, Billiards in ideal hyperbolic polygons, Discrete Contin. Dyn. Syst., 29 (2011), 893-908.  doi: 10.3934/dcds.2011.29.893.  Google Scholar

[13]

B. Cha and D. H. Kim, Lagrange spectrum of Romik's dynamical system, preprint, arXiv: 1903.02882. Google Scholar

[14]

B. Cha and D. H. Kim, Number theoretical properties of Romik's dynamical system, Bull. Korean Math. Soc., 57 (2020), 251-274.  doi: 10.4134/BKMS.b190163.  Google Scholar

[15]

B. ChaE. Nguyen and B. Tauber, Quadratic forms and their Berggren trees, J. Number Theory, 185 (2018), 218-256.  doi: 10.1016/j.jnt.2017.09.003.  Google Scholar

[16]

N. Chernov and R. Markarian, Introduction to the Ergodic Theory of Chaotic Billiards, 2$^nd$ edition, IMPA Mathematical Publications, Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2003.  Google Scholar

[17]

K. T. Conrad, Pythagorean descent, Semantic Scholar, 2007. Google Scholar

[18]

I. P. Cornfeld, S. V. Fomin and Y. G. Sinaĭ, Ergodic Theory, Vol. 245, Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4615-6927-5.  Google Scholar

[19]

A. Denjoy, Sur une fonction réelle de Minkowski, J. Math. Pures Appl., 17 (1938), 105-155.   Google Scholar

[20]

E. J. Eckert, The group of primitive Pythagorean triangles, Math. Mag., 57 (1984), 22-27.  doi: 10.1080/0025570X.1984.11977070.  Google Scholar

[21]

L. Elsner, The generalized spectral-radius theorem: An analytic-geometric proof, Linear Algebra Appl., 220 (1995), 151-159.  doi: 10.1016/0024-3795(93)00320-Y.  Google Scholar

[22]

D. Fried, Symbolic dynamics for triangle groups, Invent. Math., 125 (1996), 487-521.  doi: 10.1007/s002220050084.  Google Scholar

[23]

N. Guglielmi and M. Zennaro, Stability of linear problems: Joint spectral radius of sets of matrices, in Current Challenges in Stability Issues for Numerical Differential Equations, Vol. 2082, Lecture Notes in Math., Springer, Cham, 2014. doi: 10.1007/978-3-319-01300-8_5.  Google Scholar

[24]

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th edition, The Clarendon Press, Oxford University Press, New York, 1979.  Google Scholar

[25]

S. Isola, From infinite ergodic theory to number theory (and possibly back), Chaos Solitons Fractals, 44 (2011), 467-479.  doi: 10.1016/j.chaos.2011.01.015.  Google Scholar

[26]

O. Jenkinson and M. Pollicott, Joint spectral radius, Sturmian measures and the finiteness conjecture, Ergodic Theory Dynam. Systems, 38 (2018), 3062-3100.  doi: 10.1017/etds.2017.18.  Google Scholar

[27]

T. Jordan and T. Sahlsten, Fourier transforms of Gibbs measures for the Gauss map, Math. Ann., 364 (2016), 983-1023.  doi: 10.1007/s00208-015-1241-9.  Google Scholar

[28]

S. Katok, Fuchsian groups, geodesic flows on surfaces of constant negative curvature and symbolic coding of geodesics, in Homogeneous Flows, Moduli Spaces and Arithmetic, Vol. 10, Clay Math. Proc., Amer. Math. Soc., 2010.  Google Scholar

[29]

M. Kesseböhmer and B. O. Stratmann, A multifractal analysis for Stern-Brocot intervals, continued fractions and Diophantine growth rates, J. Reine Angew. Math., 605 (2007), 133-163.  doi: 10.1515/CRELLE.2007.029.  Google Scholar

[30]

R. Kołodziej, An infinite smooth invariant measure for some transformation of a circle, Bull. Acad. Polon. Sci. Sér. Sci. Math., 29 (1981), 549-551.   Google Scholar

[31]

J. C. Lagarias and Y. Wang, The finiteness conjecture for the generalized spectral radius of a set of matrices, Linear Algebra Appl., 214 (1995), 17-42.  doi: 10.1016/0024-3795(93)00052-2.  Google Scholar

[32]

C. Maclachlan and A. W. Reid, The Arithmetic of Hyperbolic 3-manifolds, Vol. 219, Graduate Texts in Mathematics, Springer-Verlag, New York, 2003. doi: 10.1007/978-1-4757-6720-9.  Google Scholar

[33]

A. Miller, Trees of integral triangles with given rectangular defect, Discrete Math., 313 (2013), 50-66.  doi: 10.1016/j.disc.2012.09.013.  Google Scholar

[34]

M. Misiurewicz, The result of Rafał Kołodziej, in Ergodic Theory (Sem., Les Plans-sur-Bex, 1980), Vol. 29, Monograph. Enseign. Math., Univ. Genève, Geneva, 1981.  Google Scholar

[35]

J. Morita, A transformation group of the Pythagorean numbers, Tsukuba J. Math., 10 (1986), 151-153.  doi: 10.21099/tkbjm/1496160398.  Google Scholar

[36]

U. Moschella, The de Sitter and anti-de Sitter sightseeing tour, in Einstein, 1905–2005, Vol. 47, Prog. Math. Phys., Birkhäuser, Basel, 2006. doi: 10.1007/3-7643-7436-5_4.  Google Scholar

[37]

K. Nomizu, The Lorentz-Poincaré metric on the upper half-space and its extension, Hokkaido Math. J., 11 (1982), 253-261.  doi: 10.14492/hokmj/1381757803.  Google Scholar

[38]

G. Panti, A general Lagrange theorem, Amer. Math. Monthly, 116 (2009), 70-74.  doi: 10.1080/00029890.2009.11920912.  Google Scholar

[39]

G. Panti, Slow continued fractions, transducers, and the Serret theorem, J. Number Theory, 185 (2018), 121-143.  doi: 10.1016/j.jnt.2017.08.034.  Google Scholar

[40]

A. M. Rockett and P. Szüsz, Continued Fractions, World Scientific Publishing Co., Inc., River Edge, NJ, 1992. doi: 10.1142/1725.  Google Scholar

[41]

V. A. Rohlin, Exact endomorphisms of a Lebesgue space, Izv. Akad. Nauk SSSR Ser. Mat., 25 (1961), 499-530.   Google Scholar

[42]

D. Romik, The dynamics of Pythagorean triples, Trans. Amer. Math. Soc., 360 (2008), 6045-6064.  doi: 10.1090/S0002-9947-08-04467-X.  Google Scholar

[43]

R. Salem, On some singular monotonic functions which are strictly increasing, Trans. Amer. Math. Soc., 53 (1943), 427-439.  doi: 10.1090/S0002-9947-1943-0007929-6.  Google Scholar

[44]

D. Singerman, Finitely maximal Fuchsian groups, J. London Math. Soc., 6 (1972), 29-38.  doi: 10.1112/jlms/s2-6.1.29.  Google Scholar

[45]

J. Smillie and C. Ulcigrai, Geodesic flow on the Teichmüller disk of the regular octagon, cutting sequences and octagon continued fractions maps, in Dynamical Numbers–Interplay Between Dynamical Systems and Number Theory, Vol. 532, Contempt. Math, Amer. Math. Soc., Providence, RI, 2010. doi: 10.1090/conm/532/10482.  Google Scholar

[46]

K. Takeuchi, Arithmetic triangle groups, J. Math. Soc. Japan, 29 (1977), 91-106.  doi: 10.2969/jmsj/02910091.  Google Scholar

[47]

M. Thaler, Transformations on $[0, \, 1]$ with infinite invariant measures, Israel J. Math., 46 (1983), 67-96.  doi: 10.1007/BF02760623.  Google Scholar

Figure 1.  A hint of the construction of the Romik map; the interval $ I_3 $ and its stereographic projection to $ [0,1] $ as thick lines
Figure 2.  The Romik map
Figure 3.  A unimodular billiard table and its associated factor map $ B $
Figure 4.  A typical $ \widetilde B $-orbit on the de Sitter space and its $ \arg $-image
Figure 5.  The invariant density for the map of Example 6.3
Figure 6.  A periodic orbit in a billiard table
Figure 7.  Superimposed graphs of $ B $ and $ T $, and the resulting Minkowski function
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