Entropy in Dimension One

This paper completely classifies which numbers arise as the topological entropy associated to postcritically finite self-maps of the unit interval. Specifically, a positive real number h is the topological entropy of a postcritically finite self-map of the unit interval if and only if exp(h) is an algebraic integer that is at least as large as the absolute value of any of the conjugates of exp(h); that is, if exp(h) is a weak Perron number. The postcritically finite map may be chosen to be a polynomial all of whose critical points are in the interval (0,1). This paper also proves that the weak Perron numbers are precisely the numbers that arise as exp(h), where h is the topological entropy associated to ergodic train track representatives of outer automorphisms of a free group.Comment: 38 pages, 15 figures. This paper was completed by the author before his death, and was uploaded by Dylan Thurston. A version including endnotes by John Milnor will appear in the proceedings of the Banff conference on Frontiers in Complex Dynamic

Representations of the quantum Teichmuller space, and invariants of surface diffeomorphisms

We investigate the representation theory of the polynomial core of the quantum Teichmuller space of a punctured surface S. This is a purely algebraic object, closely related to the combinatorics of the simplicial complex of ideal cell decompositions of S. Our main result is that irreducible finite-dimensional representations of this polynomial core are classified, up to finitely many choices, by group homomorphisms from the fundamental group of the surface to the isometry group of the hyperbolic 3--space. We exploit this connection between algebra and hyperbolic geometry to exhibit new invariants of diffeomorphisms of S.Comment: Revised introduction. To appear in Geometry & Topolog

Delusional Disorder

Excerpt: Delusional disorder is one among several types of psychotic disorders, all of which involve grossly impaired reality testing. The core feature of delusional disorder is one or more nonbizarre delusions that last for at least one month. These delusions involve situations that could plausibly happen in life. Apart from the direct impact of the delusion, persons with this disorder appear normal to others and are able to function adequately in everyday life. If the person has a mood episode (such as depression) while having delusions, it must be relatively brief in order to warrant the delusional disorder diagnosis. The delusions must not be directly caused by substance use or a general medical condition in order to fit the criteria for this diagnosis