335 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
Hyperbolic Structures on 3-manifolds, II: Surface groups and 3-manifolds which fiber over the circle
Geometrization theorem, fibered case: Every three-manifold that fibers over
the circle admits a geometric decomposition. Double limit theorem: for any
sequence of quasi-Fuchsian groups whose controlling pair of conformal
structures tends toward a pair of projectively measured laminations that bind
the surface, there is a convergent subsequence. This preprint also analyzes the
quasi-isometric geometry of quasi-Fuchsian 3-manifolds. This eprint is based on
a 1986 preprint, which was refereed and accepted for publication, but which I
neglected to correct and return. The referee's corrections have now been
incorporated, but it is largely the same as the 1986 version (which was a
significant revision of a 1981 version).Comment: 32 pages, 6 figures, revision of 1986 preprin
Minimal stretch maps between hyperbolic surfaces
This paper develops a theory of Lipschitz comparisons of hyperbolic surfaces
analogous to the theory of quasi-conformal comparisons. Extremal Lipschitz maps
(minimal stretch maps) and geodesics for the `Lipschitz metric' are
constructed. The extremal Lipschitz constant equals the maximum ratio of
lengths of measured laminations, which is attained with probability one on a
simple closed curve. Cataclysms are introduced, generalizing earthquakes by
permitting more violent shearing in both directions along a fault. Cataclysms
provide useful coordinates for Teichmuller space that are convenient for
computing derivatives of geometric function in Teichmuller space and measured
lamination space.Comment: 53 pages, 11 figures, version of 1986 preprin
Hyperbolic Structures on 3-manifolds, III: Deformations of 3-manifolds with incompressible boundary
This is the third in a series of papers constructing hyperbolic structures on
all Haken three-manifolds. This portion deals with the mixed case of the
deformation space for manifolds with incompressible boundary that are not
acylindrical, but are more complicated than interval bundles over surfaces.
This is a slight revision of a 1986 preprint, with a few figures added, and
slight clarifications of some of the text, but with no attempt to connect this
to later developments such as groups acting on R-trees, etc.Comment: 19 pages, 4 figure
Three-manifolds, Foliations and Circles, I
This paper investigates certain foliations of three-manifolds that are
hybrids of fibrations over the circle with foliated circle bundles over
surfaces: a 3-manifold slithers around the circle when its universal cover
fibers over the circle so that deck transformations are bundle automorphisms.
Examples include hyperbolic 3-manifolds of every possible homological type. We
show that all such foliations admit transverse pseudo-Anosov flows, and that in
the universal cover of the hyperbolic cases, the leaves limit to sphere-filling
Peano curves. The skew R-covered Anosov foliations of Sergio Fenley are
examples. We hope later to use this structure for geometrization of slithered
3-manifolds.Comment: 60 pages, 10 figure
On proof and progress in mathematics
In response to Jaffe and Quinn [math.HO/9307227], the author discusses forms
of progress in mathematics that are not captured by formal proofs of theorems,
especially in his own work in the theory of foliations and geometrization of
3-manifolds and dynamical systems.Comment: 17 pages. Abstract added in migration
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