Recursive tiling and geometry of piecewise rotations by π/7

Abstract

We study two piecewise affine maps on convex polygons, locally conjugate to a rotation by a multiple of π/7. We obtain a finite-order recursive tiling of the phase space by return map sub-domains of triangles and periodic heptagonal domains (cells), with scaling factors given by algebraic units. This tiling allows one to construct efficiently periodic orbits of arbitrary period, and to obtain a convergent sequence of coverings of the closure of the discontinuity set Γ. For every map for which such finite-order recursive tiling exists, we derive sufficient conditions for the equality of Hausdorff and box-counting dimensions, and for the existence of a finite, non-zero Hausdorff measure of . We then verify that these conditions apply to our models; we obtain an irreducible transcendental equation for the Hausdorff dimension involving fundamental units, and establish the existence of infinitely many disjoint invariant components of the residual set . We calculate numerically the asymptotic power law growth of the number of cells as a function of maximum return time, as well as the number of cells of diameter larger than a specified ϵ. In the latter case, the exponent is shown to coincide with the Hausdorff dimension.