Local multifractal analysis in metric spaces

Abstract

We study the local dimensions and local multifractal properties of measures on doubling metric spaces. Our aim is twofold. On one hand, we show that there are plenty of multifractal type measures in all metric spaces which satisfy only mild regularity conditions. On the other hand, we consider a local spectrum that can be used to gain finer information on the local behaviour of measures than its global counterpart.

Weak separation condition, Assouad dimension, and Furstenberg homogeneity

“…The next proposition shows that under the uniform finite clustering property, the lower local dimension of the projected measure can be obtained symbolically. It generalizes [17,Proposition 3.1]. If f is a function and µ is a measure, then the push-forward measure is denoted by f µ.…”

“…In general metric spaces there often are no non-trivial self-similar sets but Moran constructions occur naturally. Therefore it is justifiable to seek for results in metric spaces; see [1,2,17,29]. Secondly, working in a general setting often helps to uncover simpler proofs.…”

Local dimensions in Moran constructions

“…For these reasons we introduce the following relaxed notion: We say that the Moran construction {E i } i∈Σ is asymptotically spatially symmetric if in the definition, the assumption (4.2), is replaced by the weaker condition This notion is more useful in the sense that it is always possible to find such constructions under very general assumptions. For instance, in addition to the doubling property, it suffices to assume that X is uniformly perfect (see [22,Remark 4.3]). Recall that X is uniformly perfect, if there is η > 0 such that…”

The thermodynamic approach to multifractal analysis

“…In [1,68,118], the asymptotic behaviour captured by the level sets K α is localized: given ξ ∈ C(X ), one studies the level set K ξ = {x ∈ X | φ n (x) → ξ(x)}.…”