Dr Toby Hall

Dr Toby Hall 

Research Interests

Discrete dynamical systems, in particular the dynamics of surface homeomorphisms. Pruning theory. Train track algorithms.


Contact Information 

E-mail: tobyhall@liverpool.ac.uk
Phone: +44 151 794 4065
Fax: +44 151 794 4061

Department of Mathematical Sciences
University of Liverpool
Liverpool. L69 7ZL
United Kingdom


Recent Publications and Preprints

A full list of publications can be found here.



Software available for download

Trains: a program for computing train tracks of surface homeomorphisms

This program is an implementation of Bestvina and Handel's algorithm for determining train tracks of surface homeomorphisms (Topology 34 (1995), 109-140). The current version of the program is for Windows only.

You can download the executable and a help file containing detailed instructions (which should be placed in the same directory as the executable). There is also a pdf version of the documentation, which is not so conveniently indexed as the windows help file, but may be better for some purposes.

Dynn: a program for working with Dynnikov coordinates

This program is designed for calculations involving the action of braids on integral laminations and measured foliations on punctured disks, using Dynnikov's coordinate system. It is available as a Windows executable (to be run on the command line) only.

You can download the executable, and a PDF file which contains a very brief summary of Dynnikov's coordinates together with instructions for running the program.

Horseshoe Calculator

Here is a horseshoe calculator which calculates a range of braid type invariants of horseshoe periodic orbits.


This program, which is used for investigating pruning fronts in Smale's horseshoe, can be obtained here.


A little C++ program for calculating the left greedy form of a positive braid can be found here.