Pablo Shmerkin

Alan Turing 1.135

School of Mathematics,

Oxford Road, Manchester M13 9PL, UK

first name[dot]last name[at]manchester.ac.uk

Tel: +44 (0) 161 306 3214

Fax: +44 (0) 161 275 5819

Research Associate

Page contents:

Self-similar and self-affine fractals and multifractals. Ergodic fractal processes.
Geometric properties (projections, arithmetic sums, intersections, visible parts) of fractals of dynamical and arithmetic origin.
Dimension theory in dynamical systems, non-conformal repellers, thermodynamic formalism.
Arithmetic patterns in fractals. Connections to harmonic analysis. Sets of Kakeya and Furstenberg type.
Applications of iterated function systems. Digital control. Machine learning. Hybrid systems.

(With A. Ferguson and T. Jordan). The Hausdorff dimension of the projections of self-affine carpets. Fundamenta Mathematicae. To appear. Arxiv.
(With F. Nazarov and Y. Peres). Convolutions of Cantor measures without resonance. Israel J. Math. To appear. Arxiv.
(With J. Schmeling). On the dimension of iterated sumsets. Proceedings of the conference on fractals and applications, Tunisia, 2007. To appear. Arxiv.
(With A. Käenmäki). Overlapping self-affine sets of Kakeya type. Ergodic Theory Dynam. Systems 29 (2009), no. 3, 941--965. Arxiv.
(With Y. Peres). Resonance between Cantor sets. Ergodic Theory Dynam. Systems 29 (2009), no. 1, 201--221. Arxiv.
(With B. Solomyak). Zeros of {-1,0,1}-power series and connectedness loci for self-affine sets. Experiment. Math. 15 (2006), no. 4, 499--511. Arxiv.
Overlapping self-affine sets. Indiana Univ. Math. J. 55 (2006), no. 4, 1291--1331. Arxiv.
A modified multifractal formalism for a class of self-similar measures with overlap. Asian J. Math. 9 (2005), no. 3, 323--348. Arxiv.

Preprints:

(With M. Hochman). Local entropy averages and projections of fractal measures. Arxiv.

Papers nearing completion:

(With T. Jordan and B. Solomyak). Multifractal structure of Bernoulli convolutions and a class of self-affine measures.
(With I. Arhosalo, E. Järvenpää, M. Järvenpää and M. Rams). Visible parts of fractal percolation.

Other active projects:

(With G. Pete). Arithmetic patterns in random fractals.
(With D. Broomhead et al). Non-contracting iterated function systems and digital control.
(With J. Shapiro et al). Machine learning and iterated function systems.
(With A. Käenmäki). On the regularity of a typical self-affine set.
Mean-porous measures do not have full dimension.

Resonance between Cantor sets. Workshop on dynamical systems, Warsaw, December 2007.
Self affine sets in the plane: dimensions, overlaps and orthogonal projections. Fractal Geometry and Stochastics IV, Greifswald, September 2008.
Local entropy averages and projections of fractal measures. Pure Maths colloquium, University of St Andrews, November 2009. (This talk is intended for a general mathematical audience.)