📂Dynamics

Self-Similar Set

Definition ¹

For two sets $A, B$, if a bijection $f$ exists that satisfies $f(A) = B$, we say that the two sets $A, B$ are similar. For a set $X$, if there exists a subset $S \subset X$ that is similar to $X$, then $X$ is called a self-similar set.

Explanation

Note that the definition of ‘similar’ is somewhat arbitrary and invented by the author. The reference material uses the term ’expansion’ without a mathematical definition, but since self-similar sets are generally of interest in a geometric sense, a more complex $f$ was assumed here. Although the definition is derived from general sets rather than figures, the most common geometric candidate for $f$ would be an affine transformation that includes dilation, contraction, rotation, and translation.