• 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

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.

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. 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.  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. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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. 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.  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. 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.  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. 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.  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. MarmiP. 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. 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.  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. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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. 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.  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. 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.  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. 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.  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. MarmiP. 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)
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) $
$ \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 & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - A, 2012, 32 (7) : 2437-2451. doi: 10.3934/dcds.2012.32.2437

[3]

Pierre Arnoux, Thomas A. Schmidt. Commensurable continued fractions. Discrete & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - A, 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 & Continuous Dynamical Systems - A, 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]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

Xiongping Dai, Yu Huang, Mingqing Xiao. Realization of joint spectral radius via Ergodic theory. Electronic Research Announcements, 2011, 18: 22-30. doi: 10.3934/era.2011.18.22

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (24)
  • HTML views (42)
  • Cited by (0)

[Back to Top]