Pointwise Lower Dimension of Some Quasi-Bernoulli Measures and the Exact-dimensionality Property

The main purpose of this paper is to study the pointwise lower dimension of a measure, as a natural dual to the notion of pointwise Assouad dimension introduced by Anttila (2023), by a number of examples. This concept describes the extreme local behavior of a measure around a given point. We verify the exact dimensionality property for quasi-Bernoulli measures supported on self-conformal sets satisfying the strong separation condition. In particular, we compute the pointwise lower dimension explicitly for invariant measures with place-dependent probabilities on self-conformal sets, self-similar doubling measures satisfying the open set condition, and self-affine measures on Bedford–McMullen sponges satisfying the very strong separation condition, which complement the results by Anttila (2023).