About this video
This talk was part of the Dynamical Systems seminar at the University of Liverpool.
Abstract: A key aim in fractal geometry is to understand the dimension theory of "irregular" sets and measures. Whilst self-similar and self-affine sets (i.e. sets which are invariant under collections of contracting similarities or contracting affine maps) have been extensively studied, less attention has been given to sets invariant under nonlinear, non-conformal contractions. This is undoubtedly due to the additional challenges posed by working in the nonlinear setting.
In this talk we'll consider a class of measures in the plane, which are supported on attractors of iterated function systems consisting of nonlinear, non-conformal maps with triangular Jacobian matrices. We'll consider a notion of dimension for measures known as the L^q-spectrum, and using ideas from thermodynamic formalism we'll see how this can be calculated in our setting. As a corollary we'll also obtain the box dimension of the sets our measures are supported on. This is joint work with Kenneth Falconer and Jonathan Fraser.
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