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Matching for a family of infinite measure continued fraction transformations

The third author is supported by the NWO TOP-Grant No. 614.001.509

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  • As a natural counterpart to Nakada's $ \alpha $-continued fraction maps, we study a one-parameter family of continued fraction transformations with an indifferent fixed point. We prove that matching holds for Lebesgue almost every parameter in this family and that the exceptional set has Hausdorff dimension 1. Due to this matching property, we can construct a planar version of the natural extension for a large part of the parameter space. We use this to obtain an explicit expression for the density of the unique infinite $ \sigma $-finite absolutely continuous invariant measure and to compute the Krengel entropy, return sequence and wandering rate of the corresponding maps.

    Mathematics Subject Classification: Primary: 11K50, 37A05, 37A35, 37A40, Secondary: 11A55, 28D05, 37E05.


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  • Figure 1.  The Gauss map $ G $ and the flipped map $ R = 1-G $ in (a) and (b). The folded $ \alpha $-continued fraction map $ \hat S_\alpha $ and the flipped $ \alpha $-continued fraction map $ T_\alpha $ for $ \alpha < \frac12 $ in (c) and (e) and for $ \alpha > \frac12 $ in (d) and (f)

    Figure 4.  Numerical simulations of $ \mathcal{D}_\alpha $ for $ \alpha > \frac{1}{2}\sqrt{2} $

    Figure 2.  The transformation $ \mathcal{T}_{\alpha} $ maps areas on the top to areas on the bottom with the same color or pattern

    Figure 3.  The maps $ \mathcal T_\alpha $ for various values of $ \alpha $. Areas on the left are mapped to areas on the right with the same color or pattern

    Figure 5.  Maps $ T_\alpha $ and $ T_{\alpha'} $ that are not $ c $-isomorphic for any $ c \in \mathbb (0, \infty] $

    Table 1.  Invariant densities for $ \alpha \in \big[\frac12, \frac12 \sqrt 2 \big] $

    $ \boldsymbol{\alpha} $ Density $ \boldsymbol{f_\alpha} $
    $ [\frac12, g) $ $ \frac{1}{1-x} \mathbf 1_{[1-\alpha, \alpha]}(x) + \frac1{x(1-x)} \mathbf 1_{[\alpha, \frac{1-\alpha}{\alpha} ]} (x)+ \frac{x^2+1}{x(1-x^2)} \mathbf 1_{[\frac{1-\alpha}{\alpha},1]} (x) $
    $ [g, \frac23) $ $ \big(\frac{1}{1-x} + \frac{1}{x+\frac{1}{g-1}}\big)\mathbf 1_{[1-\alpha, \frac{2\alpha-1}{\alpha}]} (x) + \frac1{1-x} \mathbf 1_{[\frac{2\alpha-1}{\alpha}, \alpha ]} (x)+ $
    $ + \big(\frac{1}{1-x} + \frac{1}{x} -\frac{1}{x+\frac{1}{g}}\big) \mathbf 1_{[\alpha,\frac{2\alpha-1}{1-\alpha}]} (x)+ \frac{x^2+1}{x(1-x^2)} \mathbf 1_{[\frac{2\alpha-1}{1-\alpha},1]} (x) $
    $ [\frac23, \frac12 \sqrt{2}] $ $ \big(\frac{1}{1-x} + \frac{1}{x+\frac{1}{g-1}}\big)\mathbf 1_{[1-\alpha, \frac{2\alpha-1}{\alpha}]} (x) + \frac1{1-x} \mathbf 1_{[\frac{2\alpha-1}{\alpha}, \alpha ]} (x) + \big(\frac{1}{1-x} + \frac{1}{x} -\frac{1}{x+\frac{1}{g}}\big) \mathbf 1_{[\alpha,\frac{1-\alpha}{2\alpha-1}]} (x)+ $
    $ +\big(\frac{1}{1-x}+\frac{1}{x+1} -\frac{1}{x+ \frac1g} + \frac{1}{x} - \frac{1}{x+\frac{1}{g+1}}\big) \mathbf 1_{[\frac{1-\alpha}{2\alpha-1},1]} (x) $
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  • [1] J. Aaronson, An Introduction to Infinite Ergodic Theory, Mathematical Surveys and Monographs, 50, American Mathematical Society, Providence, RI, 1997. doi: 10.1090/surv/050.
    [2] P. Arnoux and T. A. Schmidt, Cross sections for geodesic flows and $\alpha$-continued fractions, Nonlinearity, 26 (2013), 711-726.  doi: 10.1088/0951-7715/26/3/711.
    [3] C. BonannoC. CarminatiS. Isola and G. Tiozzo, Dynamics of continued fractions and kneading sequences of unimodal maps, Discrete Contin. Dyn. Syst., 33 (2013), 1313-1332.  doi: 10.3934/dcds.2013.33.1313.
    [4] V. Botella-Soler, J. A. Oteo, J. Ros and P. Glendinning, Lyapunov exponent and topological entropy plateaus in piecewise linear maps, J. Phys. A, 46 (2013), 26pp. doi: 10.1088/1751-8113/46/12/125101.
    [5] H. BruinC. Carminati and C. Kalle, Matching for generalised $\beta$-transformations, Indag. Math. (N.S.), 28 (2017), 55-73.  doi: 10.1016/j.indag.2016.11.005.
    [6] H. BruinC. CarminatiS. Marmi and A. Profeti, Matching in a family of piecewise affine maps, Nonlinearity, 32 (2019), 172-208.  doi: 10.1088/1361-6544/aae935.
    [7] C. CarminatiS. Isola and G. Tiozzo, Continued fractions with $SL(2, Z)$-branches: Combinatorics and entropy, Trans. Amer. Math. Soc., 370 (2018), 4927-4973.  doi: 10.1090/tran/7109.
    [8] C. CarminatiS. MarmiA. Profeti and G. Tiozzo, The entropy of $\alpha$-continued fractions: Numerical results, Nonlinearity, 23 (2010), 2429-2456.  doi: 10.1088/0951-7715/23/10/005.
    [9] C. Carminati and G. Tiozzo, A canonical thickening of $\Bbb Q$ and the entropy of $\alpha$-continued fraction transformations, Ergodic Theory Dynam. Systems, 32 (2012), 1249-1269.  doi: 10.1017/S0143385711000447.
    [10] C. Carminati and G. Tiozzo, Tuning and plateaux for the entropy of $\alpha$-continued fractions, Nonlinearity, 26 (2013), 1049-1070.  doi: 10.1088/0951-7715/26/4/1049.
    [11] D. Cosper and M. Misiurewicz, Entropy locking, Fund. Math., 241 (2018), 83-96.  doi: 10.4064/fm330-5-2017.
    [12] K. DajaniD. HensleyC. Kraaikamp and V. Masarotto, Arithmetic and ergodic properties of 'flipped' continued fraction algorithms, Acta Arith., 153 (2012), 51-79.  doi: 10.4064/aa153-1-4.
    [13] K. Dajani and C. Kalle, Invariant measures, matching and the frequency of 0 for signed binary expansions, preprint, arXiv: 1703.06335.
    [14] K. Dajani and C. Kraaikamp, The mother of all continued fractions, Colloq. Math., 84/85 (2000), 109-123.  doi: 10.4064/cm-84/85-1-109-123.
    [15] K. DajaniC. Kraaikamp and W. Steiner, Metrical theory for $\alpha$-Rosen fractions, J. Eur. Math. Soc. (JEMS), 11 (2009), 1259-1283.  doi: 10.4171/JEMS/181.
    [16] A. Haas, Invariant measures and natural extensions, Canad. Math. Bull., 45 (2002), 97-108.  doi: 10.4153/CMB-2002-011-4.
    [17] Y. Hartono and C. Kraaikamp, On continued fractions with odd partial quotients, Rev. Roumaine Math. Pures Appl., 47 (2002), 43-62. 
    [18] C. Kalle, Isomorphisms between positive and negative $\beta$-transformations, Ergodic Theory Dynam. Systems, 34 (2014), 153-170.  doi: 10.1017/etds.2012.127.
    [19] C. Kalle and W. Steiner, Beta-expansions, natural extensions and multiple tilings associated with Pisot units, Trans. Amer. Math. Soc., 364 (2012), 2281-2318.  doi: 10.1090/S0002-9947-2012-05362-1.
    [20] 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.
    [21] C. Kraaikamp, A new class of continued fraction expansions, Acta Arith., 57 (1991), 1-39.  doi: 10.4064/aa-57-1-1-39.
    [22] C. KraaikampT. 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.
    [23] U. Krengel, Entropy of conservative transformations, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 7 (1967), 161-181.  doi: 10.1007/BF00532635.
    [24] L. Lewin, Polylogarithms and Associated Aunctions, North-Holland Publishing Co., New York-Amsterdam, 1981.
    [25] 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.
    [26] S. MarmiP. Moussa and J.-C. Yoccoz, The Brjuno functions and their regularity properties, Comm. Math. Phys., 186 (1997), 265-293.  doi: 10.1007/s002200050110.
    [27] H. Nakada, Metrical theory for a class of continued fraction transformations and their natural extensions, Tokyo J. Math., 4 (1981), 399-426.  doi: 10.3836/tjm/1270215165.
    [28] H. Nakada and R. Natsui, The non-monotonicity of the entropy of $\alpha$-continued fraction transformations, Nonlinearity, 21 (2008), 1207-1225.  doi: 10.1088/0951-7715/21/6/003.
    [29] O. Perron, Die Lehre von den Kettenbrüchen. Dritte, verbesserte und erweiterte Aufl. Bd. II. Analytisch-funktionentheoretische Kettenbrüche, B. G. Teubner Verlagsgesellschaft, Stuttgart, 1957.
    [30] V. A. Rohlin, Exact endomorphisms of a Lebesgue space, Izv. Akad. Nauk SSSR Ser. Mat., 25 (1961), 499-530. 
    [31] B. Schratzberger, $S$-expansions in dimension two, J. Théor. Nombres Bordeaux, 16 (2004), 705-732.  doi: 10.5802/jtnb.467.
    [32] C. E. Silva, On $\mu$-recurrent nonsingular endomorphisms, Israel J. Math., 61 (1988), 1-13.  doi: 10.1007/BF02776298.
    [33] C. E. Silva and P. Thieullen, The subadditive ergodic theorem and recurrence properties of Markovian transformations, J. Math. Anal. Appl., 154 (1991), 83-99.  doi: 10.1016/0022-247X(91)90072-8.
    [34] M. Thaler, Transformations on $[0, \, 1]$ with infinite invariant measures, Israel J. Math., 46 (1983), 67-96.  doi: 10.1007/BF02760623.
    [35] G. Tiozzo, The entropy of Nakada's $\alpha$-continued fractions: Analytical results, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 13 (2014), 1009-1037. 
    [36] R. Zweimüller, Ergodic structure and invariant densities of non-Markovian interval maps with indifferent fixed points, Nonlinearity, 11 (1998), 1263-1276.  doi: 10.1088/0951-7715/11/5/005.
    [37] R. Zweimüller, Ergodic properties of infinite measure-preserving interval maps with indifferent fixed points, Ergodic Theory Dynam. Systems, 20 (2000), 1519-1549.  doi: 10.1017/S0143385700000821.
    [38] R. Zweimüller, Surrey Notes on Infinite Ergodic Theory, 2009. Available from: http://mat.univie.ac.at/ zweimueller/MyPub/SurreyNotes.pdf.
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