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 and Continuous Dynamical Systems, 2020, 40 (7) : 4341-4378. doi: 10.3934/dcds.2020183
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.

[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.

[3]

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

[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.

[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.

[6]

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

[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.

[8]

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

[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.

[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.

[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.

[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.

[13]

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

[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.

[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.

[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.

[17]

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

[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.

[19]

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

[20]

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

[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.

[22]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[35]

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

[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.

[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.

[38]

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

[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.

[40]

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

[41]

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

[42]

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

[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.

[44]

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

[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.

[46]

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

[47]

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

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.

[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.

[3]

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

[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.

[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.

[6]

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

[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.

[8]

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

[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.

[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.

[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.

[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.

[13]

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

[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.

[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.

[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.

[17]

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

[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.

[19]

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

[20]

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

[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.

[22]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[35]

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

[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.

[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.

[38]

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

[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.

[40]

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

[41]

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

[42]

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

[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.

[44]

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

[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.

[46]

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

[47]

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

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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