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Billiards on pythagorean triples and their Minkowski functions

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

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  • 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.

    Mathematics Subject Classification: Primary: 37D40, 11J70.


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  • 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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