Research on Perron-Frobenius operator and fractals

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Expanding the idea of van der Corput sequences, we construct random numbers using inverse images of piecewise linear transformations. The discrepancy of this random number has deep connection with the ergodic properties of the dynamical system generated by piecewise linear transformations. For example, it is proved that if the dynamical system is mixing, then the random number is uniformly distributed. Moreover, we can deeply study the ergodic properties of the dynamical system by The spectra of thePerron-Frobenius operator. In terms of the spectra, the random number is uniformly distributed if 1 is a simple eigenvalue., and no other eigenvalues on the unit circle. The second greatest eigenvalue of the Perron-Frobenius operator is at least the reciprocal of the slope of the transformation in modulus. We proved that the random number is of low discrepancy if the second greatest eigenvalue equals its minimum in modulus. We extend this idea to construct higher dimensional low discrepancy. sequences, and we succeeded to construct two and three dimensional low discrepancy sequences.We also consider the Hausdorff dimensions from the view point of statistical mechanics, and proved it equals' a zero of the pressure. We also calculate the Hausdorff dimension of trees. The articles of these topics are now submitted.