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

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.
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

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V. Baladi, Unpublished,, (1989), (1989).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

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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

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

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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

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