13,081 research outputs found
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
When Perfect Fear Casts Out All Love : Christian Perspectives on the Assessment and Treatment of Shame
Shame reactions are often uncomfortable, even excruciating, for clients to work through in therapy. When not adequately treated, shame reactions can seriously disrupt the therapy process. This article seeks to equip Christian mental health professionals with practical strategies for effectively assessing and treating shame reactions. Theoretical perspectives on shame within a Christian context are also briefly discussed
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