Research on Perron-Frobenius operator and fractals

扩展了货车德Corput序列的思想，我们利用分段线性变换的逆像构造随机数。这种随机数的差异与分段线性变换所产生的动力系统的遍历性有着深刻的联系。例如，证明了如果动力系统是混合的，则随机数是均匀分布的。此外，通过Perron-Frobenius算子的谱，我们可以更深入地研究动力系统的遍历性。就谱而言，如果1是简单本征值，则随机数是均匀分布的。单位圆上没有其他特征值。Perron-Frobenius算子的第二大特征值至少是模变换斜率的倒数。证明了若第二大特征值等于其模的最小值，则该随机数是低偏差的。我们将这一思想推广到构造高维低偏差。序列，成功地构造了二维和三维低偏差序列，并从统计力学的角度考虑了Hausdorff维数，证明了它等于压力的零点。我们还计算了树的Hausdorff维数。现将这些专题的文章提交。

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.