Multifractal measures : from self-affine to nonlinear

Abstract

This thesis is based on three papers the author wrote during his time as a PhD student [28, 17, 33].

In Chapter 2 we study 𝐿[sup]𝑞-spectra of planar self-affine measures generated by diagonal matrices. We introduce a new technique for constructing and understanding examples based on combinatorial estimates for the exponential growth of certain split binomial sums. Using this approach we find counterexamples to a statement of Falconer and Miao from 2007 and a conjecture of Miao from 2008 concerning a closed form expression for the generalised dimensions of generic self-affine measures.

We also answer a question of Fraser from 2016 in the negative by proving that a certain natural closed form expression does not generally give the 𝐿[sup]𝑞-spectrum. As a further application we provide examples of self-affine measures whose 𝐿[sup]𝑞-spectra exhibit new types of phase transitions. Finally, we provide new non-trivial closed form bounds for the 𝐿[sup]𝑞-spectra, which in certain cases yield sharp results.

In Chapter 3 we study 𝐿[sup]𝑞-spectra of measures in the plane generated by certain nonlinear maps. In particular we study attractors of iterated function systems consisting of maps whose components are 𝐶[sup](1+α) and for which the Jacobian is a lower triangular matrix at every point subject to a natural domination condition on the entries. We calculate the 𝐿[sup]𝑞-spectrum of Bernoulli measures supported on such sets using an appropriately defined analogue of the singular value function and an appropriate pressure function.

In Chapter 4 we study a more general class of invariant measures supported on the attractors introduced in Chapter 3. These are pushforward quasi-Bernoulli measures, a class which includes the well-known class of Gibbs measures for Hölder continuous potentials. We show these measures are exact dimensional and that their exact dimensions satisfy a Ledrappier-Young formula.