American Institute of Mathematical Sciences

Advanced
August  2013, 33(8): 3365-3390. doi: 10.3934/dcds.2013.33.3365

On the mixing properties of piecewise expanding maps under composition with permutations

Nigel P. Byott 1, , Mark Holland 1, and  Yiwei Zhang 1,

1. 

College of Engineering, Mathematics and Physical Sciences, University of Exeter, North Park Road, Exeter EX4 4QF, United Kingdom, United Kingdom, United Kingdom

Received  July 2012 Revised  September 2012 Published  January 2013

We consider the effect on the mixing properties of a piecewise smooth interval map $f$ when its domain is divided into $N$ equal subintervals and $f$ is composed with a permutation of these. The case of the stretch-and-fold map $f(x)=mx \bmod 1$ for integers $m \geq 2$ is examined in detail. We give a combinatorial description of those permutations $\sigma$ for which $\sigma \circ f$ is still (topologically) mixing, and show that the proportion of such permutations tends to $1$ as $N \to \infty$. We then investigate the mixing rate of $\sigma \circ f$ (as measured by the modulus of the second largest eigenvalue of the transfer operator). In contrast to the situation for continuous time diffusive systems, we show that composition with a permutation cannot improve the mixing rate of $f$, but typically makes it worse. Under some mild assumptions on $m$ and $N$, we obtain a precise value for the worst mixing rate as $\sigma$ ranges through all permutations; this can be made arbitrarily close to $1$ as $N → ∞$ (with $m$ fixed). We illustrate the geometric distribution of the second largest eigenvalues in the complex plane for small $m$ and $N$, and propose a conjecture concerning their location in general. Finally, we give examples of other interval maps $f$ for which composition with permutations produces different behaviour than that obtained from the stretch-and-fold map.
Keywords: permutations, mixing properties, Markov maps., Transfer operator.
Mathematics Subject Classification: Primary: 37A25; Secondary: 37E05, 05E1.
Citation: Nigel P. Byott, Mark Holland, Yiwei Zhang. On the mixing properties of piecewise expanding maps under composition with permutations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3365-3390. doi: 10.3934/dcds.2013.33.3365
References:
[1]

P. Ashwin, M. Nicol and N. Kirkby, Acceleration of one-dimensional mixing by discontinous mappings,, J. Phys. A: Math. Gen., 310 (2002), 347.  doi: 10.1016/S0378-4371(02)00774-4.  Google Scholar

[2]

V. Baladi, Unpublished,, (1989), (1989).   Google Scholar

[3]

V. Baladi, S. Isola and B. Schmitt, Transfer operator for piecewise affine approximations of interval maps,, Ann. Inst. H Poincaré Phys. Théor., 62 (1995), 251.   Google Scholar

[4]

V. Baladi, "Positive Transfer Operators and Decay of Correlations," Advanced Series in Nonlinear Dynamics, 16,, World Sci. Publ., (2000).  doi: 10.1142/9789812813633.  Google Scholar

[5]

V. Baladi and M. Tsujii, Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms,, Ann. Inst. Fourier., 57 (2007), 127.   Google Scholar

[6]

G. Berkolaiko, Spectral gap of doubly stochastic matrices generated from equidistributed unitary matrices,, J. Phys. A: Math. Gen., 34 (2001).  doi: 10.1088/0305-4470/34/22/101.  Google Scholar

[7]

A. Boyarsky and P. Góra, "Laws of Chaos. Invariant Measures and Dynamical Systems in One Dimension,'', Probability and its Applications, (1997).  doi: 10.1007/978-1-4612-2024-4.  Google Scholar

[8]

C. Y. Chao, A remark on the eigenvalues of generalized circulants,, Portugal. Math., 37 (1978), 135.   Google Scholar

[9]

P. Collet and J.-P. Eckmann, Liapunov multipliers and decay of correlations in dynamical systems,, J. Stat. Phys., 115 (2004), 217.  doi: 10.1023/B:JOSS.0000019817.71073.61.  Google Scholar

[10]

M. Dellnitz, G. Froyland and S. Sertl, On the isolated spectrum of the Perron-Frobenius operator,, Nonlinearity, 13 (2000), 1171.  doi: 10.1088/0951-7715/13/4/310.  Google Scholar

[11]

M. Dellnitz and O. Junge, On the approximation of complicated dynamical behaviour,, SIAM J. Numer. Anal., 36 (1999), 491.  doi: 10.1137/S0036142996313002.  Google Scholar

[12]

D. Ž. Doković, Cyclic polygons, roots of polynomials with decreasing nonnegative coefficients, and eigenvalues of stochastic matrices,, Linear Algebra Appl., 142 (1990), 173.   Google Scholar

[13]

L. Flatto and J. C. Lagarias, The lap-counting function for linear mod one transformations I. Explicit formulas and renormalizability,, Ergod. Theor. Dyn. Syst., 16 (1996), 451.  doi: 10.1017/S0143385700008920.  Google Scholar

[14]

A. Fröhlich and M. J. Taylor, "Algebraic Number Theory," Cambridge Studies in Advanced Mathematics, 27,, Cambridge University Press, (1993).   Google Scholar

[15]

G. Froyland and M. Dellnitz, Detecting and locating near-optimal almost-invariant sets and cycles,, SIAM J. Sci. Comput., 24 (2003), 1839.  doi: 10.1137/S106482750238911X.  Google Scholar

[16]

P. Glendinning, Topological conjugation of Lorenz maps by $\beta-$transformations,, Math. Proc. Camb. Phil. Soc., 107 (1990), 401.  doi: 10.1017/S0305004100068675.  Google Scholar

[17]

S. Gouëzel, Sharp polynomial estimates for the decay of correlations,, Isr. J. Math., 139 (2004), 29.  doi: 10.1007/BF02787541.  Google Scholar

[18]

S. Gouëzel and C. Liverani, Banach spaces adapted to Anosov systems,, Ergod. Theor. Dyn. Syst., 26 (2006), 189.  doi: 10.1017/S0143385705000374.  Google Scholar

[19]

F. Hofbauer, The maximal measure for linear mod one transformation,, J. London Math. Soc. (2), 23 (1981), 92.  doi: 10.1112/jlms/s2-23.1.92.  Google Scholar

[20]

F. Hofbauer and G. Keller, Ergodic properties of invariant measures for piecewise monotone transformations,, Math. Z., 180 (1982), 119.  doi: 10.1007/BF01215004.  Google Scholar

[21]

Y. Hu, Decay of correlations for piecewise smooth maps with indifferent fixed points,, Ergod. Theor. Dyn. Syst., 24 (2004), 495.  doi: 10.1017/S0143385703000671.  Google Scholar

[22]

H. Ito, A new statement about the theorem determining the region of eigenvalues of stochastic matrices,, Linear Algebra Appl., 267 (1997), 241.   Google Scholar

[23]

F. I. Karpelevič, On the characteristic roots of matrices with nonnegative elements,, (Russian) Izvestiya Akad. Nauk SSSR Ser. Math., 15 (1951), 361.   Google Scholar

[24]

G. Keller, On the rate of convergence to equilibrium in one-dimensional systems,, Comm. Math. Phys., 96 (1984), 181.   Google Scholar

[25]

A. Lasota and M. Mackey, "Chaos, Fractals and Noise. Stochastic Aspects of Dynamics," Second edition, Applied Math Science, 97,, Springer-Verlag, (1994).   Google Scholar

[26]

C. Liverani, B. Saussol and S. Vaienti, A probabilistic approach to intermittency,, Ergod. Theor. Dyn. Syst., 19 (1999), 671.  doi: 10.1017/S0143385799133856.  Google Scholar

[27]

M. Mori, Fredholm determininant for piecewise linear transformations,, Osaka J. Math., 27 (1990), 81.   Google Scholar

[28]

M. Mori, Low discrepancy sequences generated by piecewise linear maps,, Monte Carlo Methods and Appl., 4 (1998), 141.  doi: 10.1515/mcma.1998.4.2.141.  Google Scholar

[29]

M. Mori, Mixing property and pseudo random sequences,, in, 48 (2006), 189.  doi: 10.1214/074921706000000211.  Google Scholar

[30]

H. H. Schaefer, "Banach Lattices and Positive Operator,'', Springer, (1974).   Google Scholar

[31]

A. Slomson, "An Introduction to Combinatorics,'', Chapman and Hall Mathematics Series, (1991).   Google Scholar

[32]

M. Viana, "Stochastic Dynamics of Deterministic Systems,'', Braz. Math. Colloq., 21 (1997).   Google Scholar

[33]

L.-S. Young, Recurrence times and rates of mixing,, Isr. J. Math., 110 (1999), 153.  doi: 10.1007/BF02808180.  Google Scholar

[34]

K..Zyczkowski, M. Kuś, W. Słomczyński and H.-J. Sommers, Random unistochastic matrices. Random matrix theory,, J. Phys. A: Math. Gen., 36 (2003), 3425.  doi: 10.1088/0305-4470/36/12/333.  Google Scholar

show all references

References:
[1]

P. Ashwin, M. Nicol and N. Kirkby, Acceleration of one-dimensional mixing by discontinous mappings,, J. Phys. A: Math. Gen., 310 (2002), 347.  doi: 10.1016/S0378-4371(02)00774-4.  Google Scholar

[2]

V. Baladi, Unpublished,, (1989), (1989).   Google Scholar

[3]

V. Baladi, S. Isola and B. Schmitt, Transfer operator for piecewise affine approximations of interval maps,, Ann. Inst. H Poincaré Phys. Théor., 62 (1995), 251.   Google Scholar

[4]

V. Baladi, "Positive Transfer Operators and Decay of Correlations," Advanced Series in Nonlinear Dynamics, 16,, World Sci. Publ., (2000).  doi: 10.1142/9789812813633.  Google Scholar

[5]

V. Baladi and M. Tsujii, Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms,, Ann. Inst. Fourier., 57 (2007), 127.   Google Scholar

[6]

G. Berkolaiko, Spectral gap of doubly stochastic matrices generated from equidistributed unitary matrices,, J. Phys. A: Math. Gen., 34 (2001).  doi: 10.1088/0305-4470/34/22/101.  Google Scholar

[7]

A. Boyarsky and P. Góra, "Laws of Chaos. Invariant Measures and Dynamical Systems in One Dimension,'', Probability and its Applications, (1997).  doi: 10.1007/978-1-4612-2024-4.  Google Scholar

[8]

C. Y. Chao, A remark on the eigenvalues of generalized circulants,, Portugal. Math., 37 (1978), 135.   Google Scholar

[9]

P. Collet and J.-P. Eckmann, Liapunov multipliers and decay of correlations in dynamical systems,, J. Stat. Phys., 115 (2004), 217.  doi: 10.1023/B:JOSS.0000019817.71073.61.  Google Scholar

[10]

M. Dellnitz, G. Froyland and S. Sertl, On the isolated spectrum of the Perron-Frobenius operator,, Nonlinearity, 13 (2000), 1171.  doi: 10.1088/0951-7715/13/4/310.  Google Scholar

[11]

M. Dellnitz and O. Junge, On the approximation of complicated dynamical behaviour,, SIAM J. Numer. Anal., 36 (1999), 491.  doi: 10.1137/S0036142996313002.  Google Scholar

[12]

D. Ž. Doković, Cyclic polygons, roots of polynomials with decreasing nonnegative coefficients, and eigenvalues of stochastic matrices,, Linear Algebra Appl., 142 (1990), 173.   Google Scholar

[13]

L. Flatto and J. C. Lagarias, The lap-counting function for linear mod one transformations I. Explicit formulas and renormalizability,, Ergod. Theor. Dyn. Syst., 16 (1996), 451.  doi: 10.1017/S0143385700008920.  Google Scholar

[14]

A. Fröhlich and M. J. Taylor, "Algebraic Number Theory," Cambridge Studies in Advanced Mathematics, 27,, Cambridge University Press, (1993).   Google Scholar

[15]

G. Froyland and M. Dellnitz, Detecting and locating near-optimal almost-invariant sets and cycles,, SIAM J. Sci. Comput., 24 (2003), 1839.  doi: 10.1137/S106482750238911X.  Google Scholar

[16]

P. Glendinning, Topological conjugation of Lorenz maps by $\beta-$transformations,, Math. Proc. Camb. Phil. Soc., 107 (1990), 401.  doi: 10.1017/S0305004100068675.  Google Scholar

[17]

S. Gouëzel, Sharp polynomial estimates for the decay of correlations,, Isr. J. Math., 139 (2004), 29.  doi: 10.1007/BF02787541.  Google Scholar

[18]

S. Gouëzel and C. Liverani, Banach spaces adapted to Anosov systems,, Ergod. Theor. Dyn. Syst., 26 (2006), 189.  doi: 10.1017/S0143385705000374.  Google Scholar

[19]

F. Hofbauer, The maximal measure for linear mod one transformation,, J. London Math. Soc. (2), 23 (1981), 92.  doi: 10.1112/jlms/s2-23.1.92.  Google Scholar

[20]

F. Hofbauer and G. Keller, Ergodic properties of invariant measures for piecewise monotone transformations,, Math. Z., 180 (1982), 119.  doi: 10.1007/BF01215004.  Google Scholar

[21]

Y. Hu, Decay of correlations for piecewise smooth maps with indifferent fixed points,, Ergod. Theor. Dyn. Syst., 24 (2004), 495.  doi: 10.1017/S0143385703000671.  Google Scholar

[22]

H. Ito, A new statement about the theorem determining the region of eigenvalues of stochastic matrices,, Linear Algebra Appl., 267 (1997), 241.   Google Scholar

[23]

F. I. Karpelevič, On the characteristic roots of matrices with nonnegative elements,, (Russian) Izvestiya Akad. Nauk SSSR Ser. Math., 15 (1951), 361.   Google Scholar

[24]

G. Keller, On the rate of convergence to equilibrium in one-dimensional systems,, Comm. Math. Phys., 96 (1984), 181.   Google Scholar

[25]

A. Lasota and M. Mackey, "Chaos, Fractals and Noise. Stochastic Aspects of Dynamics," Second edition, Applied Math Science, 97,, Springer-Verlag, (1994).   Google Scholar

[26]

C. Liverani, B. Saussol and S. Vaienti, A probabilistic approach to intermittency,, Ergod. Theor. Dyn. Syst., 19 (1999), 671.  doi: 10.1017/S0143385799133856.  Google Scholar

[27]

M. Mori, Fredholm determininant for piecewise linear transformations,, Osaka J. Math., 27 (1990), 81.   Google Scholar

[28]

M. Mori, Low discrepancy sequences generated by piecewise linear maps,, Monte Carlo Methods and Appl., 4 (1998), 141.  doi: 10.1515/mcma.1998.4.2.141.  Google Scholar

[29]

M. Mori, Mixing property and pseudo random sequences,, in, 48 (2006), 189.  doi: 10.1214/074921706000000211.  Google Scholar

[30]

H. H. Schaefer, "Banach Lattices and Positive Operator,'', Springer, (1974).   Google Scholar

[31]

A. Slomson, "An Introduction to Combinatorics,'', Chapman and Hall Mathematics Series, (1991).   Google Scholar

[32]

M. Viana, "Stochastic Dynamics of Deterministic Systems,'', Braz. Math. Colloq., 21 (1997).   Google Scholar

[33]

L.-S. Young, Recurrence times and rates of mixing,, Isr. J. Math., 110 (1999), 153.  doi: 10.1007/BF02808180.  Google Scholar

[34]

K..Zyczkowski, M. Kuś, W. Słomczyński and H.-J. Sommers, Random unistochastic matrices. Random matrix theory,, J. Phys. A: Math. Gen., 36 (2003), 3425.  doi: 10.1088/0305-4470/36/12/333.  Google Scholar

[1]

Marco Lenci. Uniformly expanding Markov maps of the real line: Exactness and infinite mixing. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3867-3903. doi: 10.3934/dcds.2017163
[2]

Dieter Mayer, Tobias Mühlenbruch, Fredrik Strömberg. The transfer operator for the Hecke triangle groups. Discrete & Continuous Dynamical Systems - A, 2012, 32 (7) : 2453-2484. doi: 10.3934/dcds.2012.32.2453
[3]

Mark F. Demers, Hong-Kun Zhang. Spectral analysis of the transfer operator for the Lorentz gas. Journal of Modern Dynamics, 2011, 5 (4) : 665-709. doi: 10.3934/jmd.2011.5.665
[4]

Haritha C, Nikita Agarwal. Product of expansive Markov maps with hole. Discrete & Continuous Dynamical Systems - A, 2019, 39 (10) : 5743-5774. doi: 10.3934/dcds.2019252
[5]

Ralf Spatzier, Lei Yang. Exponential mixing and smooth classification of commuting expanding maps. Journal of Modern Dynamics, 2017, 11: 263-312. doi: 10.3934/jmd.2017012
[6]

Nir Avni. Spectral and mixing properties of actions of amenable groups. Electronic Research Announcements, 2005, 11: 57-63.

[7]

Matúš Dirbák. Minimal skew products with hypertransitive or mixing properties. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1657-1674. doi: 10.3934/dcds.2012.32.1657
[8]

Damien Thomine. A spectral gap for transfer operators of piecewise expanding maps. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 917-944. doi: 10.3934/dcds.2011.30.917
[9]

Patricia Domínguez, Peter Makienko, Guillermo Sienra. Ruelle operator and transcendental entire maps. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 773-789. doi: 10.3934/dcds.2005.12.773
[10]

B. Fernandez, E. Ugalde, J. Urías. Spectrum of dimensions for Poincaré recurrences of Markov maps. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 835-849. doi: 10.3934/dcds.2002.8.835
[11]

Christoph Bandt, Helena PeÑa. Polynomial approximation of self-similar measures and the spectrum of the transfer operator. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4611-4623. doi: 10.3934/dcds.2017198
[12]

Yury Arlinskiĭ, Eduard Tsekanovskiĭ. Constant J-unitary factor and operator-valued transfer functions. Conference Publications, 2003, 2003 (Special) : 48-56. doi: 10.3934/proc.2003.2003.48
[13]

Gary Froyland, Simon Lloyd, Anthony Quas. A semi-invertible Oseledets Theorem with applications to transfer operator cocycles. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 3835-3860. doi: 10.3934/dcds.2013.33.3835
[14]

Almut Burchard, Gregory R. Chambers, Anne Dranovski. Ergodic properties of folding maps on spheres. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1183-1200. doi: 10.3934/dcds.2017049
[15]

Ian Melbourne, Dalia Terhesiu. Mixing properties for toral extensions of slowly mixing dynamical systems with finite and infinite measure. Journal of Modern Dynamics, 2018, 12: 285-313. doi: 10.3934/jmd.2018011
[16]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020217
[17]

Claudio Bonanno, Marco Lenci. Pomeau-Manneville maps are global-local mixing. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020309
[18]

Manuela Giampieri, Stefano Isola. A one-parameter family of analytic Markov maps with an intermittency transition. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 115-136. doi: 10.3934/dcds.2005.12.115
[19]

Denis Gaidashev, Tomas Johnson. Spectral properties of renormalization for area-preserving maps. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3651-3675. doi: 10.3934/dcds.2016.36.3651
[20]

M. R. S. Kulenović, J. Marcotte, O. Merino. Properties of basins of attraction for planar discrete cooperative maps. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020202

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (27)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences