Self-similarity in piecewise isometric systems

Periodicity of Certain Piecewise Affine Planar Maps

“…Extend the definition of σ naturally to words in A * by setting σ(vw) = σ(v)σ(w), where vw denotes the concatenation of v and w. Then we get the following lemma, which resembles Proposition 1 by Poggiaspalla [24]. Lemma 2.2.…”

Aperiodic orbits of piecewise rational rotations of convex polygons with recursive tiling

“…This association was discussed by Poggiaspalla [20] in the case of periodic sequences i, where the Bratteli diagram is stationary. We begin with the relevant definitions, taken from [19].…”

“…The additivity and scaling properties of ð ðq, LÞ, q, L, F Þ are immediate consequences of (22) and (23), and the extension to a measure follows from Theorem 3. The inequalities for the dimensions are a direct consequence of (20) and the monotone decreasing dependence of all transfer matrix elements.…”

“…The map is minimal and uniquely ergodic on the aperiodic orbits of each scaling sequence, and the symbolic dynamics may be described as a Vershik map on a stationary Bratteli diagram [17][18][19][20]. Moreover, the latter is characterized by both a geometric scale factor !…”

“…In the recursively tiled piecewise rotation models with quadratic irrational parameter, the ergodic and fractal properties of the aperiodic orbits are well understood [20]. The map is minimal and uniquely ergodic on the aperiodic orbits of each scaling sequence, and the symbolic dynamics may be described as a Vershik map on a stationary Bratteli diagram [17][18][19][20].…”