The Visible Part of Plane Self-similar Sets
نویسندگان
چکیده
Given a compact subset F of R2, the visible part VθF of F from direction θ is the set of x in F such that the half-line from x in direction θ intersects F only at x. It is suggested that if dimH F ≥ 1, then dimH VθF = 1 for almost all θ, where dimH denotes Hausdorff dimension. We confirm this when F is a self-similar set satisfying the convex open set condition and such that the orthogonal projection of F onto every line is an interval. In particular the underlying similarities may involve arbitrary rotations and F need not be connected.
منابع مشابه
Quantitative visibility estimates for unrectifiable sets in the plane
The “visibility” of a planar set S from a point a is defined as the normalized size of the radial projection of S from a to the unit circle centered at a. Simon and Solomyak [36] proved that unrectifiable self-similar one-sets are invisible from every point in the plane. We quantify this by giving an upper bound on the visibility of δ-neighbourhoods of such sets. We also prove lower bounds on t...
متن کاملThe random Fibonacci recurrence and the visible points of the plane
Abstract In this letter, we show a connection between the random Fibonacci recurrence and the visible points of the plane. In particular, we show that by suitably modifying the rules of the random Fibonacci map, there is a unique correspondence between the visible points (points with relative prime coordinates) of the first quadrant and the vertices of a self-similar graph (what we call the Fib...
متن کاملDigraph Self-Similar Sets and Aperiodic Tilings
Self-similarity is a concept often associated with fractal geometry. There are many interesting self-similar sets in the plane which would not generally be considered fractal, however (although their boundaries might be fractal). Such sets provide a fresh way of looking at tilings of the plane. Furthermore, a generalization of self-similarity, called digraph self-similarity, provides a way to c...
متن کاملA Flexible Link Radar Control Based on Type-2 Fuzzy Systems
An adaptive neuro fuzzy inference system based on interval Gaussian type-2 fuzzy sets in the antecedent part and Gaussian type-1 fuzzy sets as coefficients of linear combination of input variables in the consequent part is presented in this paper. The capability of the proposed method (we named ANFIS2) for function approximation and dynamical system identification is remarkable. The structure o...
متن کاملVisibility for Self-similar Sets of Dimension One in the Plane
We prove that a purely unrectifiable self-similar set of finite 1dimensional Hausdorff measure in the plane, satisfying the Open Set Condition, has radial projection of zero length from every point.
متن کامل