Dynamical behaviors of point-mass singular inner functions

Abstract

The purpose of this thesis is to make a complete study on the dynamical behavior of the singular inner function $M_{\zeta,\alpha}$ whose singular measure is the point-mass $\alpha > 0$ at $\zeta$ on the unit circle : its iterates, Denjoy-Wolff points, Julia set, ergodic proerties and computer graphics. Its(n+1)st iterate $M_{\zeta,\alpha}^{n+1}$ is also a singular function and has the integral representation. Its explicit integral representation is given in Chapter 2. The location of the Denjoy-Wolff point of $M_{\zeta,\alpha}$ is determined in Chapter 3. The location of the Denjoy-Wolff point is closely related to the Julia set of $M_{\zeta,\alpha}$ as well as to the ergodic properties of the restriction of $M_{\zeta,\alpha}$ on the unit circle $\partial\theta$. These relations are completely characterized in Chapters 4 and 5. In chapter 6, we visualize the locations of and the magnitudes of point masses of the iterate $M_{\zeta,\alpha}^{n+1}$ by circumscribing or inscribing circles. In Chapter 7, we discuss Newton``s method for finding fixed points of $M_{\zeta,\alpha}$ and prove that its Julia set is bounded. Computer-generated images are presanted for the dynamical behaviors of Newton``s method as well as for the aesthetic computer graphics.