Self-similarity is the property of an object that looks "roughly" the same on any scale. Fractals are a particularly interesting
class of self-similar objects. Self-similar objects with parameters and are described by a power law such as
For a self-similar attractor on the real line defined by contracting
similarities, Kittle and Kogler
(2026) reported that, in the absence of exact overlaps,
Harris, J. W. and Stocker, H. "Scaling Invariance and Self-Similarity" and "Construction of Self-Similar Objects." §4.11.1-4.11.2
in Handbook
of Mathematics and Computational Science. New York: Springer-Verlag, p. 113,
1998.Hutchinson, J. "Fractals and Self-Similarity." Indiana
Univ. J. Math.30, 713-747, 1981.Kittle, S. and Kogler, C.
"The Exact Overlaps Conjecture for Self-Similar Measures on the Real Line."
7 Oct 2026. https://arxiv.org/abs/2610.10511.
and 
are described by a power law such as
on the real line defined by contracting
similarities 
, Kittle and Kogler
(2026) reported that, in the absence of exact overlaps,
. Their generalized exact
overlaps conjecture gives the dimension of self-similar probability
measures in terms of composition entropy and contraction rates.
