The tangent measure distribution of self-conformal fractals (Q701809)

The tangent measure distributions of a fractal measure \(\mu\) at a point \(x\) of its support are random measures which reflect, loosely speaking, the scaled structure of \(\mu\) seen by an observer zooming into \(x\). They were first introduced in an unpublished work of Bandt, see \textit{S. Graf} [Monatsh. Math. 120, No.~3--4, 223--246 (1995; Zbl 0841.28011)], though crucial ideas were already present in the seminal work of \textit{U. Zähle} [Probab. Theory Relat. Fields 80, No.1, 79--100 (1988; Zbl 0638.60064)]. In the present paper tangent measure distributions of self-conformal fractals are investigated. It is shown that for this class of fractal measures there is a unique tangent measure distribution at \(\mu\)-almost every point \(x\), and, moreover, this tangent measure distribution is independent of \(x\).