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On the mixing properties of piecewise expanding maps under composition with permutations
In this talk, I will use a toy model (i.e., the one dimensional stretch and fold map) as an example to provide a brief introduction on the relationships between the spectral properties of the associated transfer operator and the estimations of the optimal mixing rate of the mixing process. Moreover, I will address how the optimal mixing rate varies according to the stretch and fold map has cutting and shuffling'' behavior (i.e., composing with a permutation). If time permits, I will also talk about how to interpret this problem to the eigenvalue estimations for the Random bi-stochastic matrices (free probability theory) and the locations of poles of the dynamical zeta function.