Mondays at 2pm
Frank Adams 1, Alan Turing Building
Contacts Donald Robertson Yotam Smilansky
The boundary of chaos and the boundary of positive Hausdorff dimension of survivor sets for two-branch maps of the interval
Paul Glendinning
University of Manchester
Two classical problems in bifurcation theory are the characterisation of the boundary of chaos (in the sense of positive topological entropy) of families of maps, and the boundary of positive Hausdorff dimension of survivor sets in open maps (maps with 'holes' in phase space, the survivor set is the set of points whose forward iterates never land in the hole). In one dimension these two problems are effectively equivalent, and I will give a complete description of the boundary for classes of expanding maps with two monotonic branches and either a hole or a plateau. There is a natural two-parameter family associated with these maps (the position of the hole/plateau) and I will show that there are two classes of codimension one transitions for these maps (and uncountably many codimension two points!). Much of this work is joint with Clement Hege.Jonathan Fraser
University of St Andrews
Let E be a non-empty compact subset of the Riemann sphere and T be a rational map of degree at least two. The associated orbital set is defined to be the backwards orbit of E under T. I will consider under what circumstances the upper box dimension of the orbital set may be expressed in terms of the upper box dimensions of the Julia set of T and the initial set E. Our results extend previous work on inhomogeneous iterated function systems and Kleinian orbital sets to the setting of complex dynamical systems. This is joint work with my PhD student Yunlong Xu.Christian Bönicke
Newcastle University
Eng-Jon Ong
Queen Mary University of London
Dimension drop and estimating the gap
Peej Ingarfield
University of Manchester
Iterated function systems (IFS) have played a key role in the study of fractals. When considering an IFS, studying what separation properties it has gives a great amount of information about the fractal associated to the IFS. In the case that the IFS satisfies the open set condition then Hutchinsons formula gives an exact form for the Hausdorff dimension of the fractal. This leads to the natural question of what can we say about Hausdorff dimension when an IFS fails to meet the open set condition or even has overlaps. In this talk we will discuss exact overlaps and how they are conjectured to be the cause of a phenomenon called dimension drop. After this context has been established we shall compare methods to estimate the amount this dimension can drop for a class of self similar measures with overlaps.Hyperbolic correspondences
Laurent Bartholdi
CNRS Lyon
In joint work with Dima Dudko and Kevin Pilgrim, we consider certain correspondences on a Riemann surface, and show that they admit a weak form of hyperbolicity: sufficiently long loops get shorter under lifting at a fixed point and closing. This applies in particular to the correspondences induced on Moduli space studied by Thurston, and lets us deduce: for any rational map on Cp with 4 post-critical points, there is a finite invariant collection of isotopy classes of curves into which every curve is attracted under repeated lifting. I will explain the theory behind all this: an encoding of maps and correspondences by an algebraic gadget called a "biset": a set with two commuting group actions.