American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Compactness of transfer operators and spectral representation of Ruelle zeta functions for super-continuous functions
  • DCDS Home
  • This Issue
  • Next Article
    Global existence of strong solutions to a biological network formulation model in 2+1 dimensions
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 and Continuous Dynamical Systems, 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.

[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. 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.

[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. 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.

[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.

[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.

[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.

[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. 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.

[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. 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.

[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. 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.

[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. 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.

[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.

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.

[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. 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.

[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. 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.

[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.

[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.

[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.

[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. 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.

[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. 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.

[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. 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.

[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. 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.

[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.

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 Options

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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

Download full-size image

Download as PowerPoint slide

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) $
Table Options

Download as excel

$ \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) $
[1]

Laura Luzzi, Stefano Marmi. On the entropy of Japanese continued fractions. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 673-711. doi: 10.3934/dcds.2008.20.673
[2]

Marc Kessböhmer, Bernd O. Stratmann. On the asymptotic behaviour of the Lebesgue measure of sum-level sets for continued fractions. Discrete and Continuous Dynamical Systems, 2012, 32 (7) : 2437-2451. doi: 10.3934/dcds.2012.32.2437
[3]

Pierre Arnoux, Thomas A. Schmidt. Commensurable continued fractions. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4389-4418. doi: 10.3934/dcds.2014.34.4389
[4]

Claudio Bonanno, Carlo Carminati, Stefano Isola, Giulio Tiozzo. Dynamics of continued fractions and kneading sequences of unimodal maps. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1313-1332. doi: 10.3934/dcds.2013.33.1313
[5]

Élise Janvresse, Benoît Rittaud, Thierry de la Rue. Dynamics of $\lambda$-continued fractions and $\beta$-shifts. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1477-1498. doi: 10.3934/dcds.2013.33.1477
[6]

Lulu Fang, Min Wu. Hausdorff dimension of certain sets arising in Engel continued fractions. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2375-2393. doi: 10.3934/dcds.2018098
[7]

Doug Hensley. Continued fractions, Cantor sets, Hausdorff dimension, and transfer operators and their analytic extension. Discrete and Continuous Dynamical Systems, 2012, 32 (7) : 2417-2436. doi: 10.3934/dcds.2012.32.2417
[8]

John Kieffer and En-hui Yang. Ergodic behavior of graph entropy. Electronic Research Announcements, 1997, 3: 11-16.

[9]

Kanji Inui, Hikaru Okada, Hiroki Sumi. The Hausdorff dimension function of the family of conformal iterated function systems of generalized complex continued fractions. Discrete and Continuous Dynamical Systems, 2020, 40 (2) : 753-766. doi: 10.3934/dcds.2020060
[10]

Svetlana Katok, Ilie Ugarcovici. Theory of $(a,b)$-continued fraction transformations and applications. Electronic Research Announcements, 2010, 17: 20-33. doi: 10.3934/era.2010.17.20
[11]

Oliver Jenkinson. Every ergodic measure is uniquely maximizing. Discrete and Continuous Dynamical Systems, 2006, 16 (2) : 383-392. doi: 10.3934/dcds.2006.16.383
[12]

Ryszard Rudnicki. An ergodic theory approach to chaos. Discrete and Continuous Dynamical Systems, 2015, 35 (2) : 757-770. doi: 10.3934/dcds.2015.35.757
[13]

Thierry de la Rue. An introduction to joinings in ergodic theory. Discrete and Continuous Dynamical Systems, 2006, 15 (1) : 121-142. doi: 10.3934/dcds.2006.15.121
[14]

Wenxiang Sun, Cheng Zhang. Zero entropy versus infinite entropy. Discrete and Continuous Dynamical Systems, 2011, 30 (4) : 1237-1242. doi: 10.3934/dcds.2011.30.1237
[15]

Danilo Coelho, David Pérez-Castrillo. On Marilda Sotomayor's extraordinary contribution to matching theory. Journal of Dynamics and Games, 2015, 2 (3&4) : 201-206. doi: 10.3934/jdg.2015001
[16]

Jane Hawkins, Michael Taylor. The maximal entropy measure of Fatou boundaries. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4421-4431. doi: 10.3934/dcds.2018192
[17]

Wen Huang, Leiye Xu, Shengnan Xu. Ergodic measures of intermediate entropy for affine transformations of nilmanifolds. Electronic Research Archive, 2021, 29 (4) : 2819-2827. doi: 10.3934/era.2021015
[18]

Jean Dolbeault, Giuseppe Toscani. Fast diffusion equations: Matching large time asymptotics by relative entropy methods. Kinetic and Related Models, 2011, 4 (3) : 701-716. doi: 10.3934/krm.2011.4.701
[19]

Mrinal Kanti Roychowdhury. Quantization coefficients for ergodic measures on infinite symbolic space. Discrete and Continuous Dynamical Systems, 2014, 34 (7) : 2829-2846. doi: 10.3934/dcds.2014.34.2829
[20]

Kathryn Lindsey, Rodrigo Treviño. Infinite type flat surface models of ergodic systems. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5509-5553. doi: 10.3934/dcds.2016043

2021 Impact Factor: 1.588

Tools

Metrics

  • PDF downloads (184)
  • HTML views (114)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy