The fractal structure of elliptical polynomial spirals

Abstract

We investigate fractal aspects of elliptical polynomial spirals; that is, planar spirals with differing polynomial rates of decay in the two axis directions. We give a full dimensional analysis of these spirals, computing explicitly their intermediate, box-counting and Assouad-type dimensions. An exciting feature is that these spirals exhibit two phase transitions within the Assouad spectrum, the first natural class of fractals known to have this property. We go on to use this dimensional information to obtain bounds for the Hölder regularity of maps that can deform one spiral into another, generalising the ‘winding problem’ of when spirals are bi-Lipschitz equivalent to a line segment. A novel feature is the use of fractional Brownian motion and dimension profiles to bound the Hölder exponents.

An upper bound for the intermediate dimensions of Bedford–McMullen carpets

“…For  < 1, a natural covering strategy to improve on the exponent given by the box dimension is to use covering sets with diameter of the two permissible extremes, i.e., either ı 1= or ı. It turns out that this strategy is already optimal for the case of elliptical polynomial spirals [8], for concentric spheres [25] and also for the family of countable convergent sequences [14] F p D °0; Very recent preprints [4,5] show examples where the use of more than two scales is necessary.…”

“…For example, in [7,9], continuity at  D 0 was used to relate the box dimensions of the projections of a set to the Hausdorff dimension of the set. Other applications include gaining information about the Hölder distortion of maps between sets [3,8] or deciding whether two sets are Lipschitz equivalent [4,Example 2.12].…”

Fractal analysis of degenerate spiral trajectories of a class of ordinary differential equations

“…Degenerate spirals near such singular points (hence spirals different from weak focus spirals) appear in complex swirling flows. We cite [2]: "Most naturally occurring spirals are anisotropic, developing in systems with inherent asymmetry, such as elliptical whirlpools forming in a flowing body of water. Another simple example arises in Newtonian mechanics: suppose a weight attached to an elastic band is rotated about an axis parallel to the ground.…”

“…However, if the system is allowed to decelerate, the weight will follow a spiral trajectory that will become increasingly elongated in the vertical direction as the relative contribution of gravitational force grows." (Burrell,[2], p. 1) We focus on a particular deformation (to be specified later) of X = −ny 2n−1 ∂ ∂x +mx 2m−1 ∂ ∂y , with m, n ∈ N\{0} and m+n > 2, or Y = (y−x 2n ) ∂ ∂x with n ∈ N \ {0}. The vector field X has a center at (x, y) = (0, 0), with H(x, y) = x 2m + y 2n as a first integral, while Y has the curve of singularities C = {y = x 2n } and horizontal regular orbits (see Figure 1).…”

“…For some other examples of "irregular" spiral trajectories and the calculation of their box dimensions and Minkowski content see e.g. [2,18,20,24,27,26]. In this paper we will often speak about a trajectory "near the origin (x, y) = (0, 0)".…”