Mathmatical study of complex systems

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We study the dynamics of piecewise expanding maps in higher dimension. On the contrary to the one dimensional case, we show that a C^γ piecewise expanding map does not necessarily admit an absolutely continuous invariant measure. Also we show that if we assume that the map is piecewise analytic in 2dim or piecewise affine in higher dimension, it always admits an absolutely continuous invariant measure.We also studied the so-called global lycoupled maps from a mathematical viewpoint. We made an extensive numerical experiment for 10 globally coupled logisitic map and came to the idea that the so-called "Chaotic inenerancy" in this case is caused by a hierarchy of on-off intermittencies on the invariant subspaces.