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Memory loss for time-dependent piecewise expanding systems in higher dimension

We prove a counterpart of exponential decay of correlations for certain non- stationary systems. Namely, given two probability measures absolutely continuous with respect to a reference measure, their quasi-Holder distance (and in particular their L 1 distance) decreases exponentially under action by compositions of arbitrarily chosen maps close to those that are both piecewise expanding and mixing in a certain sense. This paper studies statistical properties of time-dependent dynamical systems. In such systems, the dynamical model itself is allowed to vary with time. An important example is the flow generated by a non-autonomous vector field. Perhaps the vector field depends on physical parameters that vary with time. We address memory loss for time-dependent dynamical systems, an analog of decay of correlations. The memory loss problem has been studied extensively in the contexts of stochastic differential equations (SDEs), random dynamical systems 1 , and autonomous (time-