Angels' staircases, Sturmian sequences, and trajectories on homothety surfaces

Figure 1.  LEFT: The union of the rectangles $ R_s^\pm $. RIGHT: Edge identifications that produce the surface $ X_s $. Horizontal edges are identified by translations; vertical edges are identified by homotheties. $ X_s^+ $ and $ X_s^- $ are genus $ 1 $ subsurfaces with boundary. They are joined along the saddle connection $ E $

Figure 2.  The graph of $ \Delta_s^-(x) $ when $ s = 2/3 $, drawn only for $ 0 \le x \le 1 $. The graph of $ \Delta_s^+(x) $ appears the same; the differences occur only at the jumps, which are dense but countable

Figure 3.  "Tongues of angels:" Each tongue corresponds to a rational number in $ (0,1) $. The boundary curves are drawn using the formulas for $ \Delta_s^\pm(k/n) $

Figure 4.  The graphs of two functions $ \Upsilon_{s,\xi}^\pm(x) $ with $ s = 0.95 $, drawn only for $ 0 \le x \le 1 $. TOP: $ \xi = (1+\sqrt5)/2 $. BOTTOM: $ \xi = \pi $

Figure 5.  The image of $ \Upsilon_{s,\xi}^+ $ for $ 0<\xi<1 $

Figure 6.  Stacking diagram for the word $ BAABA $

Figure 7.  The trajectories $ \tau_\xi^- $ and $ \tau_\xi^+ $, drawn on a partial stacking diagram. LEFT: $ \xi \notin \mathbb{Q} $, as in §4.3. These trajectories are parallel and bound a strip in $ X_s $. RIGHT: $ \xi = k/n \in \mathbb{Q} $, as in §4.4. These trajectories will intersect after crossing $ k + n $ edges

Figure 8.  LEFT: A surface $ X_s^u $ with $ 0 < u < 1 $. Lower-case letters indicate dimensions. Capital letters indicate gluings between edges. RIGHT: The surface $ X_{1/2}^{1/2} $, which was the first surface the authors considered

Figure 9.  LEFT: The rectangles $ R_{s_1}^+ $ and $ R_{s_2}^- $. RIGHT: Edge identifications to form the surface $ X_{s_1,s_2} $. $ X_{s_1,s_2}^+ $ is isomorphic to $ X_{s_1}^+ $, and $ X_{s_1,s_2}^- $ is isomorphic to $ X_{s_2}^- $, as defined in §2.6