|Dec 16, 2022
The space of geodesics on a metric graph has three important invariant measures for geodesic flow that reflect the geometric, dynamical, and probabilistic properties of the metric graph. The measures are constructed by dynamical invariants and measure classes on the boundary of the universal covering tree. In this thesis, we focus on the structure of the metric graphs that determines the dynamical invariants and the boundary measure classes.
First, we formulate three boundary measure classes using potential functions analogous to the manifold cases: visibility measures, Patterson-Sullivan measures, and harmonic measures. We show that there is an edge length which is a necessary and sufficient condition to the equivalence of two of these measure classes (theorem 3.4.2, theorem 3.4.3, theorem 3.4.4).
Next, we use the dynamical invariant and boundary measures to study the brain network. Regarding the brain network as a metric graph, we compute the volume entropy and Patterson-Sullivan measure numerically. Comparing the values between the tinnitus group and the non-tinnitus group, we strengthen the tinnitus cause interpretation based on the Bayesian hypothesis and the triple network model.
We also obtain a result of topological data analysis on medical science. Using the Mapper algorithm, we represent data space as a metric graph and propose a grouping method based on the structure of the metric graph. In this framework, we find the new subtype of Miral regurgitation patients.
Finally, we improve a well-known result in the Diophantine approximation. We construct a fractal set contained in weighted singular vectors using tree structure and the shadowing property in homogeneous dynamics. Observing the tree using lattice point counting, we obtain a nontrivial lower bound of Hausdorff dimension of weighted singular vectors.