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 and Continuous Dynamical Systems, 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), 347-363. doi: 10.1016/S0378-4371(02)00774-4.

[2]

V. Baladi, Unpublished, (1989), cited in [9].

[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), 251-265.

[4]

V. Baladi, "Positive Transfer Operators and Decay of Correlations," Advanced Series in Nonlinear Dynamics, 16, World Sci. Publ., River Edge, NJ, 2000. doi: 10.1142/9789812813633.

[5]

V. Baladi and M. Tsujii, Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms, Ann. Inst. Fourier., 57 (2007), 127-154.

[6]

G. Berkolaiko, Spectral gap of doubly stochastic matrices generated from equidistributed unitary matrices, J. Phys. A: Math. Gen., 34 (2001), L319-L326. doi: 10.1088/0305-4470/34/22/101.

[7]

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

[8]

C. Y. Chao, A remark on the eigenvalues of generalized circulants, Portugal. Math., 37 (1978), 135-144.

[9]

P. Collet and J.-P. Eckmann, Liapunov multipliers and decay of correlations in dynamical systems, J. Stat. Phys., 115 (2004) 217-254. doi: 10.1023/B:JOSS.0000019817.71073.61.

[10]

M. Dellnitz, G. Froyland and S. Sertl, On the isolated spectrum of the Perron-Frobenius operator, Nonlinearity, 13 (2000), 1171-1188. doi: 10.1088/0951-7715/13/4/310.

[11]

M. Dellnitz and O. Junge, On the approximation of complicated dynamical behaviour, SIAM J. Numer. Anal., 36 (1999), 491-515. doi: 10.1137/S0036142996313002.

[12]

D. Ž. Doković, Cyclic polygons, roots of polynomials with decreasing nonnegative coefficients, and eigenvalues of stochastic matrices, Linear Algebra Appl., 142 (1990), 173-193.

[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), 451-491. doi: 10.1017/S0143385700008920.

[14]

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

[15]

G. Froyland and M. Dellnitz, Detecting and locating near-optimal almost-invariant sets and cycles, SIAM J. Sci. Comput., 24 (2003), 1839-1863. doi: 10.1137/S106482750238911X.

[16]

P. Glendinning, Topological conjugation of Lorenz maps by $\beta-$transformations, Math. Proc. Camb. Phil. Soc., 107 (1990), 401-413. doi: 10.1017/S0305004100068675.

[17]

S. Gouëzel, Sharp polynomial estimates for the decay of correlations, Isr. J. Math., 139 (2004), 29-65. doi: 10.1007/BF02787541.

[18]

S. Gouëzel and C. Liverani, Banach spaces adapted to Anosov systems, Ergod. Theor. Dyn. Syst., 26 (2006), 189-217. doi: 10.1017/S0143385705000374.

[19]

F. Hofbauer, The maximal measure for linear mod one transformation, J. London Math. Soc. (2), 23 (1981), 92-112. doi: 10.1112/jlms/s2-23.1.92.

[20]

F. Hofbauer and G. Keller, Ergodic properties of invariant measures for piecewise monotone transformations, Math. Z., 180 (1982), 119-140. doi: 10.1007/BF01215004.

[21]

Y. Hu, Decay of correlations for piecewise smooth maps with indifferent fixed points, Ergod. Theor. Dyn. Syst., 24 (2004), 495-524. doi: 10.1017/S0143385703000671.

[22]

H. Ito, A new statement about the theorem determining the region of eigenvalues of stochastic matrices, Linear Algebra Appl., 267 (1997), 241-246.

[23]

F. I. Karpelevič, On the characteristic roots of matrices with nonnegative elements, (Russian) Izvestiya Akad. Nauk SSSR Ser. Math., 15 (1951), 361-383.

[24]

G. Keller, On the rate of convergence to equilibrium in one-dimensional systems, Comm. Math. Phys., 96 (1984), 181-193.

[25]

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

[26]

C. Liverani, B. Saussol and S. Vaienti, A probabilistic approach to intermittency, Ergod. Theor. Dyn. Syst., 19 (1999), 671-685. doi: 10.1017/S0143385799133856.

[27]

M. Mori, Fredholm determininant for piecewise linear transformations, Osaka J. Math., 27 (1990), 81-116.

[28]

M. Mori, Low discrepancy sequences generated by piecewise linear maps, Monte Carlo Methods and Appl., 4 (1998), 141-162. doi: 10.1515/mcma.1998.4.2.141.

[29]

M. Mori, Mixing property and pseudo random sequences, in "Dynamics & Stochastics," IMS Lecture Notes-Monogr. Ser., 48, Inst. Math. Statist., Beachwood, OH, (2006), 189-197. doi: 10.1214/074921706000000211.

[30]

H. H. Schaefer, "Banach Lattices and Positive Operator,'' Springer, New York-Heidelberg, 1974.

[31]

A. Slomson, "An Introduction to Combinatorics,'' Chapman and Hall Mathematics Series, Chapman and Hall, Ltd., London, 1991.

[32]

M. Viana, "Stochastic Dynamics of Deterministic Systems,'' Braz. Math. Colloq., 21, IMPA, 1997.

[33]

L.-S. Young, Recurrence times and rates of mixing, Isr. J. Math., 110 (1999), 153-188. doi: 10.1007/BF02808180.

[34]

K..Zyczkowski, M. Kuś, W. Słomczyński and H.-J. Sommers, Random unistochastic matrices. Random matrix theory, J. Phys. A: Math. Gen., 36 (2003), 3425-3450. doi: 10.1088/0305-4470/36/12/333.

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), 347-363. doi: 10.1016/S0378-4371(02)00774-4.

[2]

V. Baladi, Unpublished, (1989), cited in [9].

[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), 251-265.

[4]

V. Baladi, "Positive Transfer Operators and Decay of Correlations," Advanced Series in Nonlinear Dynamics, 16, World Sci. Publ., River Edge, NJ, 2000. doi: 10.1142/9789812813633.

[5]

V. Baladi and M. Tsujii, Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms, Ann. Inst. Fourier., 57 (2007), 127-154.

[6]

G. Berkolaiko, Spectral gap of doubly stochastic matrices generated from equidistributed unitary matrices, J. Phys. A: Math. Gen., 34 (2001), L319-L326. doi: 10.1088/0305-4470/34/22/101.

[7]

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

[8]

C. Y. Chao, A remark on the eigenvalues of generalized circulants, Portugal. Math., 37 (1978), 135-144.

[9]

P. Collet and J.-P. Eckmann, Liapunov multipliers and decay of correlations in dynamical systems, J. Stat. Phys., 115 (2004) 217-254. doi: 10.1023/B:JOSS.0000019817.71073.61.

[10]

M. Dellnitz, G. Froyland and S. Sertl, On the isolated spectrum of the Perron-Frobenius operator, Nonlinearity, 13 (2000), 1171-1188. doi: 10.1088/0951-7715/13/4/310.

[11]

M. Dellnitz and O. Junge, On the approximation of complicated dynamical behaviour, SIAM J. Numer. Anal., 36 (1999), 491-515. doi: 10.1137/S0036142996313002.

[12]

D. Ž. Doković, Cyclic polygons, roots of polynomials with decreasing nonnegative coefficients, and eigenvalues of stochastic matrices, Linear Algebra Appl., 142 (1990), 173-193.

[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), 451-491. doi: 10.1017/S0143385700008920.

[14]

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

[15]

G. Froyland and M. Dellnitz, Detecting and locating near-optimal almost-invariant sets and cycles, SIAM J. Sci. Comput., 24 (2003), 1839-1863. doi: 10.1137/S106482750238911X.

[16]

P. Glendinning, Topological conjugation of Lorenz maps by $\beta-$transformations, Math. Proc. Camb. Phil. Soc., 107 (1990), 401-413. doi: 10.1017/S0305004100068675.

[17]

S. Gouëzel, Sharp polynomial estimates for the decay of correlations, Isr. J. Math., 139 (2004), 29-65. doi: 10.1007/BF02787541.

[18]

S. Gouëzel and C. Liverani, Banach spaces adapted to Anosov systems, Ergod. Theor. Dyn. Syst., 26 (2006), 189-217. doi: 10.1017/S0143385705000374.

[19]

F. Hofbauer, The maximal measure for linear mod one transformation, J. London Math. Soc. (2), 23 (1981), 92-112. doi: 10.1112/jlms/s2-23.1.92.

[20]

F. Hofbauer and G. Keller, Ergodic properties of invariant measures for piecewise monotone transformations, Math. Z., 180 (1982), 119-140. doi: 10.1007/BF01215004.

[21]

Y. Hu, Decay of correlations for piecewise smooth maps with indifferent fixed points, Ergod. Theor. Dyn. Syst., 24 (2004), 495-524. doi: 10.1017/S0143385703000671.

[22]

H. Ito, A new statement about the theorem determining the region of eigenvalues of stochastic matrices, Linear Algebra Appl., 267 (1997), 241-246.

[23]

F. I. Karpelevič, On the characteristic roots of matrices with nonnegative elements, (Russian) Izvestiya Akad. Nauk SSSR Ser. Math., 15 (1951), 361-383.

[24]

G. Keller, On the rate of convergence to equilibrium in one-dimensional systems, Comm. Math. Phys., 96 (1984), 181-193.

[25]

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

[26]

C. Liverani, B. Saussol and S. Vaienti, A probabilistic approach to intermittency, Ergod. Theor. Dyn. Syst., 19 (1999), 671-685. doi: 10.1017/S0143385799133856.

[27]

M. Mori, Fredholm determininant for piecewise linear transformations, Osaka J. Math., 27 (1990), 81-116.

[28]

M. Mori, Low discrepancy sequences generated by piecewise linear maps, Monte Carlo Methods and Appl., 4 (1998), 141-162. doi: 10.1515/mcma.1998.4.2.141.

[29]

M. Mori, Mixing property and pseudo random sequences, in "Dynamics & Stochastics," IMS Lecture Notes-Monogr. Ser., 48, Inst. Math. Statist., Beachwood, OH, (2006), 189-197. doi: 10.1214/074921706000000211.

[30]

H. H. Schaefer, "Banach Lattices and Positive Operator,'' Springer, New York-Heidelberg, 1974.

[31]

A. Slomson, "An Introduction to Combinatorics,'' Chapman and Hall Mathematics Series, Chapman and Hall, Ltd., London, 1991.

[32]

M. Viana, "Stochastic Dynamics of Deterministic Systems,'' Braz. Math. Colloq., 21, IMPA, 1997.

[33]

L.-S. Young, Recurrence times and rates of mixing, Isr. J. Math., 110 (1999), 153-188. doi: 10.1007/BF02808180.

[34]

K..Zyczkowski, M. Kuś, W. Słomczyński and H.-J. Sommers, Random unistochastic matrices. Random matrix theory, J. Phys. A: Math. Gen., 36 (2003), 3425-3450. doi: 10.1088/0305-4470/36/12/333.

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