# American Institute of Mathematical Sciences

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
##### References:

show all references

##### References:
 [1] M. R. S. Kulenović, J. Marcotte, O. Merino. Properties of basins of attraction for planar discrete cooperative maps. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2721-2737. doi: 10.3934/dcdsb.2020202 [2] Fei Liu, Xiaokai Liu, Fang Wang. On the mixing and Bernoulli properties for geodesic flows on rank 1 manifolds without focal points. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021057 [3] Dor Elimelech. Permutations with restricted movement. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021038 [4] Ajay Jasra, Kody J. H. Law, Yaxian Xu. Markov chain simulation for multilevel Monte Carlo. Foundations of Data Science, 2021, 3 (1) : 27-47. doi: 10.3934/fods.2021004 [5] Monica Conti, Lorenzo Liverani, Vittorino Pata. A note on the energy transfer in coupled differential systems. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021042 [6] Peter H. van der Kamp, David I. McLaren, G. R. W. Quispel. Generalised Manin transformations and QRT maps. Journal of Computational Dynamics, 2021  doi: 10.3934/jcd.2021009 [7] Yila Bai, Haiqing Zhao, Xu Zhang, Enmin Feng, Zhijun Li. The model of heat transfer of the arctic snow-ice layer in summer and numerical simulation. Journal of Industrial & Management Optimization, 2005, 1 (3) : 405-414. doi: 10.3934/jimo.2005.1.405 [8] A. Kochergin. Well-approximable angles and mixing for flows on T^2 with nonsingular fixed points. Electronic Research Announcements, 2004, 10: 113-121. [9] Carlos Gutierrez, Nguyen Van Chau. A remark on an eigenvalue condition for the global injectivity of differentiable maps of $R^2$. Discrete & Continuous Dynamical Systems, 2007, 17 (2) : 397-402. doi: 10.3934/dcds.2007.17.397 [10] Liviana Palmisano, Bertuel Tangue Ndawa. A phase transition for circle maps with a flat spot and different critical exponents. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021067 [11] Bo Tan, Qinglong Zhou. Approximation properties of Lüroth expansions. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2873-2890. doi: 10.3934/dcds.2020389 [12] Craig Cowan. Supercritical elliptic problems involving a Cordes like operator. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021037 [13] John Villavert. On problems with weighted elliptic operator and general growth nonlinearities. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021023 [14] Kehan Si, Zhenda Xu, Ka Fai Cedric Yiu, Xun Li. Open-loop solvability for mean-field stochastic linear quadratic optimal control problems of Markov regime-switching system. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021074 [15] Shu-Yu Hsu. Existence and properties of ancient solutions of the Yamabe flow. Discrete & Continuous Dynamical Systems, 2018, 38 (1) : 91-129. doi: 10.3934/dcds.2018005 [16] Fumihiko Nakamura. Asymptotic behavior of non-expanding piecewise linear maps in the presence of random noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2457-2473. doi: 10.3934/dcdsb.2018055 [17] Claudia Lederman, Noemi Wolanski. An optimization problem with volume constraint for an inhomogeneous operator with nonstandard growth. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2907-2946. doi: 10.3934/dcds.2020391 [18] Markus Harju, Jaakko Kultima, Valery Serov, Teemu Tyni. Two-dimensional inverse scattering for quasi-linear biharmonic operator. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021026 [19] Zhaoqiang Ge. Controllability and observability of stochastic implicit systems and stochastic GE-evolution operator. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021009 [20] Zhang Chao, Minghua Yang. BMO type space associated with Neumann operator and application to a class of parabolic equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021104

2019 Impact Factor: 1.338