10.48617/ETD.23
Liu, Qing
Qing
Liu
Dynamics and Rigidity of the Morse Boundary
Brandeis University
2022
Doctoral thesis
dynamics
hyperbolic space
Morse boundary
quasi-conformal
Abbott, Carolyn
Carolyn
Abbott
0000-0001-5899-2661
57190747629
Rafi, Kasra
Kasra
Rafi
Charney, Ruth
Ruth
Charney
0000-0001-8239-1436
en
application/pdf
Open
Let $X$ be a proper geodesic metric space and let $G$ be a group of isometries of $X$ which acts geometrically. Cordes constructed the Morse boundary of $X$ which generalizes the contracting boundary for CAT(0) spaces and the visual boundary for hyperbolic spaces. For the first part, we characterize Morse elements in $G$ by their fixed points on the Morse boundary $\partial_MX$. The dynamics on the Morse boundary is very similar to that of a $\delta$-hyperbolic space. In particular, we show that the action of $G$ on $\partial_MX$ is minimal if $G$ is not virtually cyclic. We also get a uniform convergence result on the Morse boundary which gives us a weak north-south dynamics for a Morse isometry. This generalizes the work of Murray in the case of the contracting boundary of a CAT(0) space.For the second part, we investigate additional structures on the Morse boundary which determine the space up to a quasi-isometry. We prove that a homeomorphism between the Morse boundaries of two proper, cocompact spaces is induced by a quasi-isometry if and only if both the homeomorphism and its inverse are bih\"{o}lder, or quasisymmetric, or strongly quasi-conformal.