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CS - Dr Ian Morris

Thermodynamic formalism of graph-directed self-affine sets

Supervisor: Dr Ian Morris

Project description:

Standard examples of fractal sets such as the Sierpinski triangle, von Koch curve and Menger sponge have the property of being self-similar, which means that they are equal to the union of finitely many re-scaled exact copies of themselves. 

The fractal geometry of self-similar sets is well studied and is in many respects very well-understood. A self-affine set, on the other hand, has the property of being equal to finitely many linearly distorted copies of itself, and such sets have only started to be well understood in the last decade.   

The objective of this project is to transfer some recent results on self-affine sets into the related domain of graph-directed self-affine sets. The main goal will be to establish the thermodynamic formalism of graph-directed self-affine sets. This allows high-dimensional measures on these sets to be constructed which in turn give sharp estimates on the fractal set’s Hausdorff dimension. Subsequently, this work could be extended to study projections or sum sets of certain types of self-affine fractal.  

Applicants for this project should already have some experience of fractal geometry and measure theory. Prior knowledge of ergodic theory would be useful but is not strictly essential. 

References:

  • Jairo Bochi and Ian Morris, Equilibrium states of generalised singular value potentials and applications to affine iterated function systems, 2018. 
  • Antti Käenmäki and Ian Morris, Structure of equilibrium states on self-affine sets and strict monotonicity of affinity dimension, 2018.  
  • Ábel Farkas, Projections of self-similar sets with no separation condition, 2016. 
  • De-Jun Feng and Antti Käenmäki, Equilibrium states of the pressure function for products of matrices, 2011. 

Funding Notes:

This project is open to candidates applying for CSC/Underrepresented Studentships and self-funded candidates.

Further information: 
How to apply 
Entry requirements 
Fees and funding

PhD Information Session 2026:
On Wednesday 14 January, we will be holding a short information session about PhD studies in Mathematics at QMUL. For full details about the event, please visit: https://www.qmul.ac.uk/maths/postgraduate/postgraduate-research/phd-information-session-2026/

 

As one of the UK’s most diverse universities, QMUL fosters an inclusive and supportive academic community.

The School of Mathematical Sciences is committed to the equality of opportunities and to advancing women’s careers. As holders of a Bronze Athena SWAN award, we offer family-friendly benefits and support part-time study.  

 

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