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, 2013, 33 (8) : 3365-3390. doi: 10.3934/dcds.2013.33.3365
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show all references

References:
[1]

J. Phys. A: Math. Gen., 310 (2002), 347-363. doi: 10.1016/S0378-4371(02)00774-4.  Google Scholar

[2]

(1989), cited in [9]. Google Scholar

[3]

Ann. Inst. H Poincaré Phys. Théor., 62 (1995), 251-265.  Google Scholar

[4]

World Sci. Publ., River Edge, NJ, 2000. doi: 10.1142/9789812813633.  Google Scholar

[5]

Ann. Inst. Fourier., 57 (2007), 127-154.  Google Scholar

[6]

J. Phys. A: Math. Gen., 34 (2001), L319-L326. doi: 10.1088/0305-4470/34/22/101.  Google Scholar

[7]

Probability and its Applications, Birkhäuser Boston, Inc., Boston, 1997. doi: 10.1007/978-1-4612-2024-4.  Google Scholar

[8]

Portugal. Math., 37 (1978), 135-144.  Google Scholar

[9]

J. Stat. Phys., 115 (2004) 217-254. doi: 10.1023/B:JOSS.0000019817.71073.61.  Google Scholar

[10]

Nonlinearity, 13 (2000), 1171-1188. doi: 10.1088/0951-7715/13/4/310.  Google Scholar

[11]

SIAM J. Numer. Anal., 36 (1999), 491-515. doi: 10.1137/S0036142996313002.  Google Scholar

[12]

Linear Algebra Appl., 142 (1990), 173-193. Google Scholar

[13]

Ergod. Theor. Dyn. Syst., 16 (1996), 451-491. doi: 10.1017/S0143385700008920.  Google Scholar

[14]

Cambridge University Press, Cambridge, 1993.  Google Scholar

[15]

SIAM J. Sci. Comput., 24 (2003), 1839-1863. doi: 10.1137/S106482750238911X.  Google Scholar

[16]

Math. Proc. Camb. Phil. Soc., 107 (1990), 401-413. doi: 10.1017/S0305004100068675.  Google Scholar

[17]

Isr. J. Math., 139 (2004), 29-65. doi: 10.1007/BF02787541.  Google Scholar

[18]

Ergod. Theor. Dyn. Syst., 26 (2006), 189-217. doi: 10.1017/S0143385705000374.  Google Scholar

[19]

J. London Math. Soc. (2), 23 (1981), 92-112. doi: 10.1112/jlms/s2-23.1.92.  Google Scholar

[20]

Math. Z., 180 (1982), 119-140. doi: 10.1007/BF01215004.  Google Scholar

[21]

Ergod. Theor. Dyn. Syst., 24 (2004), 495-524. doi: 10.1017/S0143385703000671.  Google Scholar

[22]

Linear Algebra Appl., 267 (1997), 241-246.  Google Scholar

[23]

(Russian) Izvestiya Akad. Nauk SSSR Ser. Math., 15 (1951), 361-383.  Google Scholar

[24]

Comm. Math. Phys., 96 (1984), 181-193.  Google Scholar

[25]

Springer-Verlag, New York, 1994.  Google Scholar

[26]

Ergod. Theor. Dyn. Syst., 19 (1999), 671-685. doi: 10.1017/S0143385799133856.  Google Scholar

[27]

Osaka J. Math., 27 (1990), 81-116.  Google Scholar

[28]

Monte Carlo Methods and Appl., 4 (1998), 141-162. doi: 10.1515/mcma.1998.4.2.141.  Google Scholar

[29]

in "Dynamics & Stochastics," IMS Lecture Notes-Monogr. Ser., 48, Inst. Math. Statist., Beachwood, OH, (2006), 189-197. doi: 10.1214/074921706000000211.  Google Scholar

[30]

Springer, New York-Heidelberg, 1974.  Google Scholar

[31]

Chapman and Hall Mathematics Series, Chapman and Hall, Ltd., London, 1991.  Google Scholar

[32]

Braz. Math. Colloq., 21, IMPA, 1997. Google Scholar

[33]

Isr. J. Math., 110 (1999), 153-188. doi: 10.1007/BF02808180.  Google Scholar

[34]

J. Phys. A: Math. Gen., 36 (2003), 3425-3450. doi: 10.1088/0305-4470/36/12/333.  Google Scholar

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