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

Abstract

We study piecewise rational rotations of convex polygons with a recursive tiling property. For these dynamical systems, the set AE of discontinuity-avoiding aperiodic orbits decomposes into invariant subsets endowed with a hierarchical symbolic dynamics (Vershik map on a Bratteli diagram). Under conditions which guarantee a form of asymptotic temporal scaling, we prove minimality and unique ergodicity for each invariant component. We study the multi-fractal properties of the model with respect to recurrence times, deriving a method of successive approximations for the generalized dimensions (q). We consider explicit examples in which the trace of the rotation matrix is a quadratic or cubic irrational, and evaluate numerically, with high precision, the function (q) and its Legendre transform.

Renormalization of polygon exchange maps arising from corner percolation

“…Another example of a renormalizable piecewise rotation is provided in [LKV04]. And in [Low07], a general theory of renormalization of piecewise rotations is developed. In all these cases, periodic points are shown to be of full measure in the dynamical system.…”

Approach to a rational rotation number in a piecewise isometric system

“…The symmetry properties of the atoms Λ out m for m ≥ 3 are proved with an argument analogous to that used in the proof of theorem 6. Specifically, in place of (26) and (27), we have the symmetric points…”

“…For rational rotations, the stable regions in phase space -the ellipses of figure 1-become convex polygons, and the system parameters (such as the quantity λ in (1)) are algebraic numbers. Much of recent research has been devoted to quadratic parameter values, plus some scattered results for cubic parameters [18,21,26,27,30]. Rational rotation numbers with prime denominator were considered in [17] from a ring-theoretic angle, in a rather general setting.…”

The Triple Lattice PETs

“…If we restrict the isometry on each piece to either a translation or a rotation by qπ for q ∈ Q, we call the map T a piecewise rational rotation. (See [2], [8], [14] and [13] for references.) There is a natural construction of PETs from piecewise rational rotations.…”

“…A classical example of renormalizable piecewise isometries is described in the survey paper [7] by Goetz. In the paper [13], Lowenstein develops a general theory of piecewise rational rotations. Hooper gives the first example of PETs in 2-dimensional parameter space which is invariant under renormalization [11].…”