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November  2020, 40(11): 6309-6330. doi: 10.3934/dcds.2020281

Matching for a family of infinite measure continued fraction transformations

Charlene Kalle 1, , Niels Langeveld 1, , Marta Maggioni 1, and  Sara Munday 2,

1. 

Department of Mathematics, Leiden University, Niels Bohrweg 1, 2333CA Leiden, The Netherlands
2. 

John Cabot University, 00165 Roma, Italy, Via della Lungara 233, 00165 Roma, Italy

Received  December 2019 Revised  May 2020 Published  July 2020

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

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.

Keywords: Continued fractions, invariant measure, Krengel entropy, infinite ergodic theory, matching.
Mathematics Subject Classification: Primary: 11K50, 37A05, 37A35, 37A40, Secondary: 11A55, 28D05, 37E05.
Citation: Charlene Kalle, Niels Langeveld, Marta Maggioni, Sara Munday. Matching for a family of infinite measure continued fraction transformations. Discrete & Continuous Dynamical Systems - A, 2020, 40 (11) : 6309-6330. doi: 10.3934/dcds.2020281
References:
[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.  Google Scholar

[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.  Google Scholar

[3]

C. Bonanno, C. Carminati, S. 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.  Google Scholar

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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.  Google Scholar

[5]

H. Bruin, C. 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.  Google Scholar

[6]

H. Bruin, C. Carminati, S. Marmi and A. Profeti, Matching in a family of piecewise affine maps, Nonlinearity, 32 (2019), 172-208.  doi: 10.1088/1361-6544/aae935.  Google Scholar

[7]

C. Carminati, S. 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.  Google Scholar

[8]

C. Carminati, S. Marmi, A. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

D. Cosper and M. Misiurewicz, Entropy locking, Fund. Math., 241 (2018), 83-96.  doi: 10.4064/fm330-5-2017.  Google Scholar

[12]

K. Dajani, D. Hensley, C. 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.  Google Scholar

[13]

K. Dajani and C. Kalle, Invariant measures, matching and the frequency of 0 for signed binary expansions, preprint, arXiv: 1703.06335. Google Scholar

[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.  Google Scholar

[15]

K. Dajani, C. Kraaikamp and W. Steiner, Metrical theory for $\alpha$-Rosen fractions, J. Eur. Math. Soc. (JEMS), 11 (2009), 1259-1283.  doi: 10.4171/JEMS/181.  Google Scholar

[16]

A. Haas, Invariant measures and natural extensions, Canad. Math. Bull., 45 (2002), 97-108.  doi: 10.4153/CMB-2002-011-4.  Google Scholar

[17]

Y. Hartono and C. Kraaikamp, On continued fractions with odd partial quotients, Rev. Roumaine Math. Pures Appl., 47 (2002), 43-62.   Google Scholar

[18]

C. Kalle, Isomorphisms between positive and negative $\beta$-transformations, Ergodic Theory Dynam. Systems, 34 (2014), 153-170.  doi: 10.1017/etds.2012.127.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

C. Kraaikamp, A new class of continued fraction expansions, Acta Arith., 57 (1991), 1-39.  doi: 10.4064/aa-57-1-1-39.  Google Scholar

[22]

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.  Google Scholar

[23]

U. Krengel, Entropy of conservative transformations, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 7 (1967), 161-181.  doi: 10.1007/BF00532635.  Google Scholar

[24]

L. Lewin, Polylogarithms and Associated Aunctions, North-Holland Publishing Co., New York-Amsterdam, 1981.  Google Scholar

[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.  Google Scholar

[26]

S. Marmi, P. Moussa and J.-C. Yoccoz, The Brjuno functions and their regularity properties, Comm. Math. Phys., 186 (1997), 265-293.  doi: 10.1007/s002200050110.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

O. Perron, Die Lehre von den Kettenbrüchen. Dritte, verbesserte und erweiterte Aufl. Bd. II. Analytisch-funktionentheoretische Kettenbrüche, B. G. Teubner Verlagsgesellschaft, Stuttgart, 1957.  Google Scholar

[30]

V. A. Rohlin, Exact endomorphisms of a Lebesgue space, Izv. Akad. Nauk SSSR Ser. Mat., 25 (1961), 499-530.   Google Scholar

[31]

B. Schratzberger, $S$-expansions in dimension two, J. Théor. Nombres Bordeaux, 16 (2004), 705-732.  doi: 10.5802/jtnb.467.  Google Scholar

[32]

C. E. Silva, On $\mu$-recurrent nonsingular endomorphisms, Israel J. Math., 61 (1988), 1-13.  doi: 10.1007/BF02776298.  Google Scholar

[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.  Google Scholar

[34]

M. Thaler, Transformations on $[0, \, 1]$ with infinite invariant measures, Israel J. Math., 46 (1983), 67-96.  doi: 10.1007/BF02760623.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

R. Zweimüller, Surrey Notes on Infinite Ergodic Theory, 2009. Available from: http://mat.univie.ac.at/ zweimueller/MyPub/SurreyNotes.pdf. Google Scholar

show all references

References:
[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.  Google Scholar

[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.  Google Scholar

[3]

C. Bonanno, C. Carminati, S. 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.  Google Scholar

[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.  Google Scholar

[5]

H. Bruin, C. 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.  Google Scholar

[6]

H. Bruin, C. Carminati, S. Marmi and A. Profeti, Matching in a family of piecewise affine maps, Nonlinearity, 32 (2019), 172-208.  doi: 10.1088/1361-6544/aae935.  Google Scholar

[7]

C. Carminati, S. 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.  Google Scholar

[8]

C. Carminati, S. Marmi, A. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

D. Cosper and M. Misiurewicz, Entropy locking, Fund. Math., 241 (2018), 83-96.  doi: 10.4064/fm330-5-2017.  Google Scholar

[12]

K. Dajani, D. Hensley, C. 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.  Google Scholar

[13]

K. Dajani and C. Kalle, Invariant measures, matching and the frequency of 0 for signed binary expansions, preprint, arXiv: 1703.06335. Google Scholar

[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.  Google Scholar

[15]

K. Dajani, C. Kraaikamp and W. Steiner, Metrical theory for $\alpha$-Rosen fractions, J. Eur. Math. Soc. (JEMS), 11 (2009), 1259-1283.  doi: 10.4171/JEMS/181.  Google Scholar

[16]

A. Haas, Invariant measures and natural extensions, Canad. Math. Bull., 45 (2002), 97-108.  doi: 10.4153/CMB-2002-011-4.  Google Scholar

[17]

Y. Hartono and C. Kraaikamp, On continued fractions with odd partial quotients, Rev. Roumaine Math. Pures Appl., 47 (2002), 43-62.   Google Scholar

[18]

C. Kalle, Isomorphisms between positive and negative $\beta$-transformations, Ergodic Theory Dynam. Systems, 34 (2014), 153-170.  doi: 10.1017/etds.2012.127.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

C. Kraaikamp, A new class of continued fraction expansions, Acta Arith., 57 (1991), 1-39.  doi: 10.4064/aa-57-1-1-39.  Google Scholar

[22]

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.  Google Scholar

[23]

U. Krengel, Entropy of conservative transformations, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 7 (1967), 161-181.  doi: 10.1007/BF00532635.  Google Scholar

[24]

L. Lewin, Polylogarithms and Associated Aunctions, North-Holland Publishing Co., New York-Amsterdam, 1981.  Google Scholar

[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.  Google Scholar

[26]

S. Marmi, P. Moussa and J.-C. Yoccoz, The Brjuno functions and their regularity properties, Comm. Math. Phys., 186 (1997), 265-293.  doi: 10.1007/s002200050110.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

O. Perron, Die Lehre von den Kettenbrüchen. Dritte, verbesserte und erweiterte Aufl. Bd. II. Analytisch-funktionentheoretische Kettenbrüche, B. G. Teubner Verlagsgesellschaft, Stuttgart, 1957.  Google Scholar

[30]

V. A. Rohlin, Exact endomorphisms of a Lebesgue space, Izv. Akad. Nauk SSSR Ser. Mat., 25 (1961), 499-530.   Google Scholar

[31]

B. Schratzberger, $S$-expansions in dimension two, J. Théor. Nombres Bordeaux, 16 (2004), 705-732.  doi: 10.5802/jtnb.467.  Google Scholar

[32]

C. E. Silva, On $\mu$-recurrent nonsingular endomorphisms, Israel J. Math., 61 (1988), 1-13.  doi: 10.1007/BF02776298.  Google Scholar

[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.  Google Scholar

[34]

M. Thaler, Transformations on $[0, \, 1]$ with infinite invariant measures, Israel J. Math., 46 (1983), 67-96.  doi: 10.1007/BF02760623.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

R. Zweimüller, Surrey Notes on Infinite Ergodic Theory, 2009. Available from: http://mat.univie.ac.at/ zweimueller/MyPub/SurreyNotes.pdf. Google Scholar

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)
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Figure 4.  Numerical simulations of $ \mathcal{D}_\alpha $ for $ \alpha > \frac{1}{2}\sqrt{2} $
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Figure 2.  The transformation $ \mathcal{T}_{\alpha} $ maps areas on the top to areas on the bottom with the same color or pattern
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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
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Figure 5.  Maps $ T_\alpha $ and $ T_{\alpha'} $ that are not $ c $-isomorphic for any $ c \in \mathbb (0, \infty] $
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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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$ \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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