A slow triangle map with a segment of indifferent fixed points and a complete tree of rational pairs
Tóm tắt
We study the two-dimensional continued fraction algorithm introduced in Garrity (J Number Theory 88(1):86–103, 2001) and the associated triangle map T, defined on a triangle
$$\triangle \subseteq \mathbb {R}^2$$
. We introduce a slow version of the triangle map, the map S, which is ergodic with respect to the Lebesgue measure and preserves an infinite Lebesgue-absolutely continuous invariant measure. We discuss the properties that the two maps T and S share with the classical Gauss and Farey maps on the interval, including an analogue of the weak law of large numbers and of Khinchin’s weak law for the digits of the triangle sequence, the expansion associated to T. Finally, we confirm the role of the map S as a two-dimensional version of the Farey map by introducing a complete tree of rational pairs, constructed using the inverse branches of S, in the same way as the Farey tree is generated by the Farey map, and then, equivalently, generated by a generalised mediant operation.
Tài liệu tham khảo
Aaronson, J.: “An Introduction to Infinite Ergodic Theory”. Mathematical Surveys and Monographs, vol. 50. American Mathematical Society, Providence (1997)
Amburg, I., et al.: Stern sequences for a family of multidimensional continued fractions: TRIP-Stern sequences. J. Integer Seq. 20(1), Article 17.1.7 (2017)
Bingham, N.H., Goldie, C.M., Teugels, J.L.: “Regular Variation”, Encyclopedia of Mathematics and Its Applications, vol. 27. Cambridge University Press, Cambridge (1989)
Bonanno, C., Del Vigna, A.: Representation and coding of rational pairs on a triangular tree and Diophantine approximation in \({\mathbb{R}}^{2}\). arXiv:2007.05958 [math.NT]
Bonanno, C., Isola, S.: Orderings of the rationals and dynamical systems. Colloq. Math. 116(2), 165–189 (2009)
Brentjes, A.J.: “Multidimensional Continued Fraction Algorithms”. Mathematical Centre Tracts, vol. 145. Mathematisch Centrum, Amsterdam (1981)
Garrity, T.: On periodic sequences for algebraic numbers. J. Number Theory 88(1), 86–103 (2001)
Garrity, T.: On Gauss–Kuzmin statistics and the transfer operator for a multidimensional continued fraction algorithms: the triangle map. arXiv: 1509.01840v1 [math.NT]
Garrity, T., Mcdonald, P.: Generalizing the Minkowski question mark function to a family of multidimensional continued fractions. Int. J. Number Theory 14(9), 2473–2516 (2018)
Iosifescu, M., Kraaikamp, C.: “Metrical Theory of Continued Fractions”. Mathematics and Its Applications, vol. 547. Kluwer Academic Publishers, Dordrecht (2002)
Isola, S.: From infinite ergodic theory to number theory (and possibly back). Chaos Solitons Fractals 44(7), 467–479 (2011)
Kesseböhmer, M., Munday, S., Stratmann, B.O.: Infinite Ergodic Theory of Numbers. De Gruyter Graduate. De Gruyter, Berlin (2016)
Kesseböhmer, M., Stratmann, B.O.: Fractal analysis for sets of non-differentiability of Minkowski’s question mark function. J. Number Theory 128(9), 2663–2686 (2008)
Lenci, M.: On infinite-volume mixing. Commun. Math. Phys. 298(2), 485–514 (2010)
Lenci, M.: Exactness, K-property and infinite mixing. Publ. Mat. Urug. 14, 159–170 (2013)
Lenci, M., Munday, S.: Pointwise convergence of Birkhoff averages for global observables. Chaos 28(8), 083111 (2018)
Messaoudi, A., Nogueira, A., Schweiger, F.: Ergodic properties of triangle partitions. Monatsh. Math. 157(3), 283–299 (2009)
Minkowski, H.: Geometrie der Zahlen, Gesammelte Abhandlungen, vol. 2, 1911; reprinted by Chelsea, New York, pp. 43–52 (1967)
Miao, J.J., Munday, S.: Derivatives of slippery Devil’s staircases. Discrete Contin. Dyn. Syst. Ser. S 10(2), 353–365 (2017)
Munday, S.: On the derivative of the \(\alpha \)-Farey–Minkowski function. Discrete Contin. Dyn. Syst. 34(2), 709–732 (2014)
Nakada, H., Natsui, R.: On the metrical theory of continued fraction mixing fibred systems and its application to Jacobi–Perron algorithm. Monatsh. Math. 138(4), 267–288 (2003)
Panti, G.: Multidimensional continued fractions and a Minkowski function. Monatsh. Math. 154(3), 247–264 (2008)
Schweiger, F.: Kuzmin’s theory revisited. Ergodic Theory Dyn. Syst. 20(2), 557–565 (2000)
Schweiger, F.: Multidimensional Continued Fractions. Oxford University Press, Oxford (2000)
Veech, W.A.: Interval exchange transformations. J. Anal. Math. 33, 222–272 (1978)
Vernon, R.P.: Relationships between Fibonacci-type sequences and Golden-type ratios. Notes Number Theory Discrete Math. 24(2), 85–89 (2018)
Zweimuller, R.: Surrey notes on infinite ergodic theory. http://mat.univie.ac.at/%7Ezweimueller/MyPub/SurreyNotes.pdf