Pointwise Assouad dimension for measures

On pointwise dimension of non-hyperbolic measures

We construct a diffeomorphism preserving a non-hyperbolic measure whose pointwise dimension does not exist almost everywhere. In the one-dimensional case we also show that such diffeomorphisms are typical in certain situations.

Pointwise Dimension for Regular Hyperbolic Measures for Endomorphisms

Conformal Assouad Dimension and Modulus

Let α ≥ 1 and let (X, d, μ) be an α-homogeneous metric measure space with conformal Assouad dimension equal to α. Then there exists a weak tangent of (X, d, μ) with uniformly big 1-modulus.

Assouad Dimension of Self-affine Carpets

We calculate the Assouad dimension of the self-affine carpets of Bedford and McMullen, and of Lalley and Gatzouras. We also calculate the conformal Assouad dimension of those carpets that are not self-similar.

On the Assouad dimension of self-similar sets with overlaps

It is known that, unlike the Hausdorff dimension, the Assouad dimension of a self-similar set can exceed the similarity dimension if there are overlaps in the construction. Our main result is the following precise dichotomy for self-similar sets in the line: either the weak separation property is satisfied, in which case the Hausdorff and Assouad dimensions coincide; or the weak separation prop...