# American Institute of Mathematical Sciences

2020, 16: 109-153. doi: 10.3934/jmd.2020005

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

 1 Natural Science Division, Pepperdine University, 24255 Pacific Coast Highway, Malibu, CA 90263, USA 2 Department of Mathematics, Kidder Hall 368, Oregon State University, Corvallis, OR 97331, USA

Received  September 15, 2018 Revised  February 13, 2020

A homothety surface can be assembled from polygons by identifying their edges in pairs via homotheties, which are compositions of translation and scaling. We consider linear trajectories on a $1$-parameter family of genus-$2$ homothety surfaces. The closure of a trajectory on each of these surfaces always has Hausdorff dimension $1$, and contains either a closed loop or a lamination with Cantor cross-section. Trajectories have cutting sequences that are either eventually periodic or eventually Sturmian. Although no two of these surfaces are affinely equivalent, their linear trajectories can be related directly to those on the square torus, and thence to each other, by means of explicit functions. We also briefly examine two related families of surfaces and show that the above behaviors can be mixed; for instance, the closure of a linear trajectory can contain both a closed loop and a lamination.

Citation: Joshua P. Bowman, Slade Sanderson. Angels' staircases, Sturmian sequences, and trajectories on homothety surfaces. Journal of Modern Dynamics, 2020, 16: 109-153. doi: 10.3934/jmd.2020005
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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 $E$
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
"Tongues of angels:" Each tongue corresponds to a rational number in $(0,1)$. The boundary curves are drawn using the formulas for $\Delta_s^\pm(k/n)$
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: $\xi = \pi$
The image of $\Upsilon_{s,\xi}^+$ for $0<\xi<1$
Stacking diagram for the word $BAABA$
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
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
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
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