Invariant measures of the topological flow and measures at infinity on hyperbolic groups

Figure 1.  A part of a strictly invariant family of multicones of index $ 1 $ for $ ( \Gamma, S) $, where $ \Gamma $ is a genus two surface group and $ S $ is the set of translation generators. The arrows indicate the destinations of intervals after translation by a single generator $ s \in S $ (bottom), and an expanded picture shows a small portion of the disk bounded by the thick arc (top)

Figure 2.  The histograms (the number of steps $ n = 10^3 $ and the number of samples $ 10^4 $) for the distributions of angles in $ [-3.14\dots, 3.14\dots] $ of $ \rho(w_n).o $ where $ o $ is the center of the unit disk: $ \{w_n\}_{n = 0, 1, \dots} $ is associated with the Markov chain on the strongly connected component in the underlying directed graph in an automatic structure for $ ( \Gamma, S) $ (left), and $ \{w_n\}_{n = 0, 1, \dots} $ is the simple random walk on the Cayley graph for $ ( \Gamma, S) $ (right)