Abstract
This paper is devoted to studying the boundary behavior of self-affine sets. We prove that the boundary of an integral self-affine set has Lebesgue measure zero. In addition, we consider the variety of the boundary of a self-affine set when some other contractive maps are added. We show that the complexity of the boundary of the new self-affine set may be the same, more complex, or simpler; any one of the three cases is possible.
Highlights
Let (X, ρ) be a complete matric space
If we associate the iterated function system (IFS) with a set of probability weights {pi > 0 : i = 1, . . . , m}, there exists a unique probability measure μ supported on K satisfying the equation m μ (⋅) = ∑pjμ (Sj−1 (⋅))
We consider the Lebesgue measures of the boundaries of integral self-affine sets. We prove that they have Lebesgue measure zero
Summary
Let (X, ρ) be a complete matric space. Recall that a map S : X → X is contractive if there exists a constant 0 < r < 1 such that ρ(S(x), S(y)) ≤ rρ(x, y). We call K the invariant set or attractor of the IFS. There is no method to compute the Hausdorff dimension and the Lebesgue measure L(∂K) of ∂K for overlapping self-affine set. We consider the Lebesgue measures of the boundaries of integral self-affine sets.
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
