We prove that an iterated function system of similarities on $\mathbb{R}$ satisfies the weak separation condition and has interval as its self-similar set stronger generalized finite type condition. It is unknown if assumption necessary.

Abstract We show that self-similar measures on $\mathbb R^d$ satisfying the weak separation condition are uniformly scaling. Our approach combines elementary ergodic theory with geometric analysis of structure given by condition.

We prove that any iterated function system of finite type possesses the weak separation property.

For any self-similar measure μ on R satisfying the weak separation condition, we show that there exists an open ball U0 with μ(U0) > 0 such that the distribution of μ, restricted on U0, is controlled by the products of a family of non-negative matrices, and hence μ|U0 satisfies a kind of quasi-product property. Furthermore, the multifractal formalism for μ|U0 is valid on the whole range of dime...