Research on Spectra of Perron-Frobenius operator generated by dynamical systm and random numbers

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We have studied the spectra of the Perron-Frobenius operator associated with dynamical systems. The method is to define the generating function associated with the dynamical systems, then construct renewal equations on generating functions.Then we can determine a matrix which we call Fredholm matrix. The zeros of the determinant of this matrix determine the ergodic property of the dynamical system, such as ergodicity, mixingity and decay rate of correlations of dynamical systems.Using this method, we could study rotaion numbers of dynamical system or large deviations. We can even calculate the Hausdorff dimension of fractals generated by dynamical systems.Especially, we have studied the discrepancy of random numbers generated by dynamical systems. For one dimensional cases, we can construct a theorem how to determine the discrepancy of random numbers. Using this theorem, we can determine the low discrepancy sequences generated by one dimensional dynamical systems.Recently, the progress of mathematical finance and so on, we need the higher dimensional low discrepancy sequences. However, for higher dimensional cases, we have only constructed abstract theorem to determine the spectra of the Perron-Frobenius operateor. Therefore, we have tried examples of low discrepancy sequences, and get two and three dimensional cases.