Natural extensions and entropy of $ \alpha $-continued fraction expansion maps with odd partial quotients

Yusuf Hartono; Cor Kraaikamp; Niels Langeveld; Claire Merriman

doi:10.3934/dcds.2023032

Article Contents

2023, Volume 43, Issue 8: 2852-2888. Doi: 10.3934/dcds.2023032

This issue Previous Article Next Article Concentration results for a singularly perturbed elliptic system with variable coefficients

Natural extensions and entropy of $ \alpha $-continued fraction expansion maps with odd partial quotients

1.
Department of Mathematics Education, Sriwijaya University (Unsri), Jalan Raya Palembang-Prabumulih Km 32 Indralaya 30662, Indonesia
2.
Delft University of Technology, EWI (DIAM), Mekelweg 4, 2628 CD Delft, the Netherlands
3.
Leiden University, MI, PO Box 9512, 2399RA Leiden, the Netherlands, Montan University Leoben, Franz Josef-Straße 18, 8700 Leoben, Austria
4.
Davidson College, Box 7129, Davidson, NC 28035, USA

^*Corresponding author: Yusuf Hartono
^*Corresponding author: Yusuf Hartono

Received: August 2021

Revised: December 2022

Early access: April 2023

Published: August 2023

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Abstract

In [1], Boca and the fourth author of this paper introduced a new class of continued fraction expansions with odd partial quotients, parameterized by a parameter $ \alpha\in [g, G] $, where $ g = \tfrac{1}{2}(\sqrt{5}-1) $ and $ G = g+1 = 1/g $ are the two golden mean numbers. Using operations called singularizations and insertions on the partial quotients of the odd continued fraction expansions under consideration, the natural extensions from [1] are obtained, and it is shown that for each $ \alpha, \alpha^*\in [g, G] $ the natural extensions from [1] are metrically isomorphic. An immediate consequence of this is, that the entropy of all these natural extensions is equal for $ \alpha\in [g, G] $, a fact already observed in [1]. Furthermore, it is shown that this approach can be extended to values of $ \alpha $ smaller than $ g $, and that for values of $ \alpha \in [\tfrac{1}{6}(\sqrt{13}-1), g] $ all natural extensions are still isomorphic. In the final section of this paper further attention is given to the entropy, as function of $ \alpha\in [0, G] $. It is shown that if there exists an ergodic, absolutely continuous $ T_{\alpha} $-invariant measure, in any neighborhood of $ 0 $ we can find intervals on which the entropy is decreasing, intervals on which the entropy is increasing and intervals on which the entropy is constant. Moreover, we identify the largest interval on which the entropy is constant. In order to prove this we use a phenomenon called matching.

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Mathematics Subject Classification: Primary: 28D05; Secondary: 11K50.

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Figure 1. The map $ T_\alpha $ for $ \alpha = 0.72 $

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Figure 2. The domains $ \Omega $ and $ \Omega_1 $

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Figure 3. The domains $ \Omega_1 $ and $ \Omega_\alpha $ for $ \alpha\in[g, 1) $

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Figure 4. The domain $ \Omega_G $

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Figure 5. The domain $ \Omega_G $ and the region $ \Omega_{\alpha, 1} $

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Figure 6. The regions $ \Omega_{\alpha, 1} $ and $ \Omega_{\alpha, 2} $

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Figure 7. The domain $ \Omega_\alpha $ for $ \alpha\in (1, G) $

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Figure 8. The domain $ \Omega_{g} $

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Figure 9. The domain $ \Omega_{g} $ and the region $ \Omega_{\alpha, 1} $

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Figure 10. The left part of the natural extension after "one more round" for $ \alpha\in [\frac{\sqrt{13}-1}{6} , g) $

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Figure 11. The entropy plotted as a function of $ \alpha $. On the left for the Odd $ \alpha $-continued fractions, on the right for Nakada's $ \alpha $-continued fractions. The values in these plots are obtained by simulations

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Figure 12. Simulations of the entropy as a function of $ \alpha $ on the interval $ [\frac{1}{4}, \frac{1}{3}] $. On the left for the odd $ \alpha $-continued fractions, on the right for Nakada's $ \alpha $-continued fractions

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Figure 13. Simulations of the entropy as a function of $ \alpha $ on an interval containing $ \frac{14}{47} $. On the left for the odd $ \alpha $-continued fractions, on the right for Nakada's $ \alpha $-continued fractions

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Figure 14. Left: $ \alpha<r_{2n} $, right: $ \alpha>r_{2n} $. Note that in both cases the ordering of $ \alpha $ and $ r_{2n} $ swaps but the ordering of $ \alpha-2 $ and $ r_{2n}-2 $ remains preserved

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Figure 15. The matching exponents on the largest not-covered intervals with the matching exponents $ (N, N) $ on the $ y $-axis. The bottom right picture is the interval from $ g^2 $ to the left end point of the interval $ I_3 $ for which we took rationals with denominator smaller than $ 100.000 $, the other pictures are for rationals with denominator smaller than $ 10.000 $

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Table 1. The digits of the odd $ \alpha $-continued fractions expansions of $ \alpha $ and $ \alpha-2 $ for $ \alpha $ from the sequences of (38), but without the signs

$ \alpha $	digits of the odd $ \alpha $-CF of $ \alpha $	digits of the odd $ \alpha $-CF of $ \alpha-2 $
$ a_n $	$ n $ $ n+1, 1 $	$ (1, 3, 3)^{\frac{n-1}{2}}, 1 $ $ (1, 3, 3)^{\frac{n-2}{2}}, 1, 3 $	odd even
$ b_n $	$ n+2, 1, 5, (1, 3, 3)^{\frac{n-1}{2}}, 1 $ $ n+1, (3, 3, 1)^{\frac{n}{2}} $	$ (1, 3, 3)^{\frac{n-1}{2}}, 1, 5, 1, n+2 $ $ (1, 3, 3)^{\frac{n}{2}}, n+1 $	odd even
$ c_n $	$ n+2, (1, 3, 3)^{\frac{n-1}{2}}, 3, 1, 3, 3, 1 $ $ n+1, 3, (1, 3, 3)^{\frac{n-2}{2}}, 3, 1, 3, 3, 1 $	$ (1, 3, 3)^{\frac{n-1}{2}}, 1, n+2, 5 $ $ (1, 3, 3)^{\frac{n-2}{2}}, 1, 3, n+1, 5 $	odd even
$ d_n $	$ n+2, (1, 3, 3)^{\frac{n-1}{2}}, 1 $ $ n+1, 3, (1, 3, 3)^{\frac{n-2}{2}}, 1 $	$ (1, 3, 3)^{\frac{n-1}{2}}, 1, n+2 $ $ (1, 3, 3)^{\frac{n-2}{2}}, 1, 3, n+1 $	odd even

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Table 2. $ M_{\alpha, \alpha, N} $ and $ M_{\alpha, \alpha-2, M} $ for the sequences from (38). These are found by calculating the convergents using Table 1

$ \alpha $	$ M_{\alpha, \alpha, N} $	$ M_{\alpha, \alpha-2, M} $
$ a_n $	$ \left[ \begin{array}{ c c } 0&1\\1&n \end{array} \right] $	$ \left[ \begin{array}{ c c } 4-4n&1-2n\\2n-1&n \end{array} \right] $	odd
	$ \left[ \begin{array}{ c c } 1&1\\n+1&n \end{array} \right] $	$ \left[ \begin{array}{ c c } 3-2n&1-2n\\n-1&n \end{array} \right] $	even
$ b_n $	$ \left[ \begin{array}{ c c } 4n&2n+1\\4n^2+2n+1&2n^2+2n+1 \end{array} \right] $	$ \left[ \begin{array}{ c c } -4n&-4n^2-2n-1\\2n+1&2n^2+2n+1 \end{array} \right] $	odd and even
$ c_n $	$ \left[ \begin{array}{ c c } 9n-2&5n-1\\9n^2-2n+9&n(5n-1)+5 \end{array} \right] $	$ \left[ \begin{array}{ c c } -2n^2+n-2&7n-10n^2-11\\n^2+1&n(5n-1)+5 \end{array} \right] $	odd and even
$ d_n $	$ \left[ \begin{array}{ c c } 2n-1&n\\2n^2-n+2&n^2+1 \end{array} \right] $	$ \left[ \begin{array}{ c c } 1-2n&n-2n^2-2\\n&n^2+1 \end{array} \right] $	odd and even

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References

[1]	F. P. Boca and C. Merriman, $\alpha$-expansions with odd partial quotients, J. Number Theory, 199 (2019), 322-341. doi: 10.1016/j.jnt.2018.11.015.
[2]	R. M. Burton, C. Kraaikamp and T. A. Schmidt, Natural extensions for the Rosen fractions, Trans. Amer. Math. Soc., 352 (2000), 1277-1298. doi: 10.1090/S0002-9947-99-02442-3.
[3]	K. Calta, C. Kraaikamp and T. A. Schmidt, Synchronization is full measure for all $\alpha$-deformations of an infinite class of continued fractions, Ann. Sc. Norm. Super. Pisa Cl. Sci., 20 (2020), 951-1008.
[4]	C. Carminati, S. Isola and G. Tiozzo, Continued fractions with $SL(2, \mathbb{Z})$-branches: Combinatorics and entropy, Trans. Amer. Math. Soc., 370 (2018), 4927-4973. doi: 10.1090/tran/7109.
[5]	C. Carminati, N. Langeveld and W. Steiner, Tanaka-Ito $\alpha$-continued fractions and matching, 2020., Nonlinearity, 34 (2021), 3565-3582. doi: 10.1088/1361-6544/abef75.
[6]	C. Carminati and G. Tiozzo, A canonical thickening of $\mathbb{Q}$ and the entropy of $\alpha$-continued fraction transformations, Ergodic Theory Dynam. Systems, 32 (2012), 1249-1269. doi: 10.1017/S0143385711000447.
[7]	K. Dajani and C. Kalle, A First Course in Ergodic Theory, Chapman and Hall/CRC, 2021.
[8]	K. Dajani and C. Kraaikamp, Ergodic Theory of Numbers, Carus Mathematical Monographs, 29. Mathematical Association of America, Washington, DC, 2002.
[9]	Y. Hartono and C. Kraaikamp, On continued fractions with odd partial quotients, Rev. Roum. Math. Pures Appl., 47 (2002), 43-62.
[10]	M. Iosifescu and C. Kraaikamp, Metrical Theory of Continued Fractions, Mathematics and its Applications, vol. 547, Kluwer, Dordrecht, 2002. doi: 10.1007/978-94-015-9940-5.
[11]	J. de Jonge, C. Kraaikamp and H. Nakada, Orbits of N-expansions with a finite set of digits, Monatsh. Math., 198 (2022), 79-119. doi: 10.1007/s00605-021-01658-x.
[12]	S. Katok and I. Ugarcovici, Structure of attractors for $(a, b)$-continued fraction transformations, J. Mod. Dyn., 4 (2010), 637-691. doi: 10.3934/jmd.2010.4.637.
[13]	S. Katok and I. Ugarcovici, Theory of $(a, b)$-continued fraction transformations and applications, Electron. Res. Announc.Math. Sci., 17 (2010), 20-33. doi: 10.3934/era.2010.17.20.
[14]	C. Kraaikamp, A new class of continued fraction expansions, Acta Arith., 57 (1991), 1-39. doi: 10.4064/aa-57-1-1-39.
[15]	C. Kraaikamp, T. A. Schmidt and I. Smeets, Natural extensions for $\alpha$-Rosen continued fractions, J. Math. Soc. Japan, 62 (2010), 649-671.
[16]	C. Kraaikamp, T. A. Schmidt and W. Steiner, Natural extensions and entropy of $\alpha$-continued fractions, Nonlinearity, 25 (2012), 2207-2243. doi: 10.1088/0951-7715/25/8/2207.
[17]	N. Langeveld, Matching, entropy, holes and expansions, PhD thesis, Leiden Universtity, 2019, https://hdl.handle.net/1887/81488.
[18]	L. Luzzi and S. Marmi, On the entropy of Japanese continued fractions, Discrete Contin. Dyn. Syst., 20 (2008), 673-711. doi: 10.3934/dcds.2008.20.673.
[19]	P. Moussa, A. Cassa and S. Marmi, Continued fractions and Brjuno functions, Continued fractions and geometric function theory (CONFUN) (Trondheim, 1997), J. Comput. Appl. Math., 105 (1999), 403-415. doi: 10.1016/S0377-0427(99)00029-1.
[20]	H. Nakada, Metrical theory for a class of continued fraction transformations and their natural extentions, Tokyo J. Math., 4 (1981), 399-426. doi: 10.3836/tjm/1270215165.
[21]	H. Nakada and R. Natsui, The non-monotonicity of the entropy of $\alpha$-continued fraction transformations, Nonlinearity, 6 (2008), 1207-1225. doi: 10.1088/0951-7715/21/6/003.
[22]	G. J. Rieger, On the metrical theory of continued fractions with odd partial quotients, Topics in Classical Number Theory, 1/2, (Budapest, 1981), 1371-1418; Colloq. Math. Soc. János Bolyai, North-Holland, Amsterdam-New York, 34 (1984).
[23]	V. A. Rohlin, Exact endomorphisms of a Lebesgue space, Izv. Akad. Nauk SSSR Ser. Mat., 25 (1961), 499-530.
[24]	D. Rosen, A class of continued fractions associated with certain properly discontinuous groups, Duke Math. J., 21 (1954), 549-563.
[25]	F. Schweiger, Continued fractions with odd and even partial quotients, Arbeitbericht Mathematisches Institut Salzburg, 4 (1982), 59-70.
[26]	F. Schweiger, On the approximation by continued fractions with odd and even partial quotients, Arbeitbericht Mathematisches Institut Salzburg, 1/2 (1984), 105-114.
[27]	G. I. Sebe, Gauss' problem for the continued fraction with odd partial quotients, Rev. Roumaine Math. Pures Appl., 46 (2001), 839-852.
[28]	G. I. Sebe, On convergence rate in the Gauss-Kuzmin problem for grotesque continued fractions, Monatsh. Math., 133 (2001), 241-254. doi: 10.1007/s006050170022.
[29]	S. Tanaka and S. Ito, On a family of continued fraction transformations and their ergodic properties, Tokyo J. Math., 4 (1981), 153-175. doi: 10.3836/tjm/1270215745.