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

  • *Corresponding author: Yusuf Hartono

    *Corresponding author: Yusuf Hartono 
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  • 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.

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

    Citation:

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

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

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

    Figure 4.  The domain $ \Omega_G $

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

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

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

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

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

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

    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

    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

    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

    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

    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 $

    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
     | Show Table
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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
     | Show Table
    DownLoad: CSV
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