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33
The symplectic nature of fundamental groups of surfaces
 ADV. MATH
, 1984
"... A symplectic structure on a manifold is a closed nondegenerate exterior 2form. The most common type of symplectic structure arises on a complex manifold as the imaginary part of a Hermitian metric which is Klhler. Many moduli spaces associated with Riemann surfaces have such Kahler structures: the ..."
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Cited by 223 (8 self)
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A symplectic structure on a manifold is a closed nondegenerate exterior 2form. The most common type of symplectic structure arises on a complex manifold as the imaginary part of a Hermitian metric which is Klhler. Many moduli spaces associated with Riemann surfaces have such Kahler structures: the Jacobi variety, Teichmüller space, moduli spaces of stable vector bundles and even the first real cohomology group have such structures. In all of these examples the topology of the associated spaces depends, remarkably, only on the topology of the Riemann surface, while often their complex structures vary as the complex structure of the Riemann surface changes. However, the symplectic structure of these spaces depends only on the underlying topological surface. The purpose of this paper is to present a general explanation for this phenomenon. We present a single construction which unifies all of the above examples and interprets their symplectic structures in terms of the intersection pairing on the surface. Our setup is as follows. Consider a closed oriented topological surface S with fundamental group rt and let G be a connected Lie group. The space Hom(n, G) consisting of representations rr + G (given the compactopen topology) is a real analytic variety (which is an algebraic variety if G is an algebraic group). There is a canonical Gaction on Hom(rr, G) obtained by composing representations with inner autormorphisms of G. The resulting quotient space Hom(x, G)/G is a space canonically associated with S (or 7~) and G. When G is an abelian group Hom(n, G)/G = Hom(7c, G) = H’(S; G) has a natural (abelian) group structure. The present paper addresses the question of what sort of natural structure Hom(n, G)/G possesses when G is not necessarily abelian. We find that under fairly general conditions on G (e.g., if it is reductive) Hom(r, G)/G admits in a natural way a symplectic structure which generalizes the Kahler forms on all of the spaces mentioned above.
Regular or stochastic dynamics in real analytic families of unimodal maps
, 2003
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Asymptotic Rigidity of Scaling ratios for critical circle mappings
 J. EUR. MATH. SOC. (JEMS
, 1996
"... Let f be a smooth homeomorphism of the circle having one cubicexponent critical point and irrational rotation number of bounded combinatorial type. Using certain pullback and quasiconformal surgery techniques, we prove that the scaling ratios of f about the critical point are asymptotically inde ..."
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Cited by 24 (1 self)
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Let f be a smooth homeomorphism of the circle having one cubicexponent critical point and irrational rotation number of bounded combinatorial type. Using certain pullback and quasiconformal surgery techniques, we prove that the scaling ratios of f about the critical point are asymptotically independent of f. This settles in particular the golden mean universality conjecture. We introduce the notion of holomorphic commuting pair, a complex dynamical system that, in the analytic case, represents an extension of f to the complex plane and behaves somewhat as a quadraticlike mapping. We define a suitable renormalization operator that acts on such objects. Through careful analysis of the family of entire mappings given by z ↦ → z+θ − 1 sin 2πz, θ real, we construct examples of holomorphic commuting 2π pairs, from which certain necessary limit set prerigidity results are extracted. The rigidity problem for f is thereby reduced to one of renormalization convergence. We handle this last problem by means of Teichmüller extremal methods made available through the recent work of Sullivan on Riemann surface laminations and renormalization of unimodal mappings.
RIGIDITY OF ESCAPING DYNAMICS FOR TRANSCENDENTAL ENTIRE FUNCTIONS
, 2008
"... We prove an analog of Böttcher’s theorem for transcendental entire functions in the EremenkoLyubich class B. More precisely, let f and g be entire functions with bounded sets of singular values and suppose that f and g belong to the same parameter space (i.e., are quasiconformally equivalent in th ..."
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Cited by 18 (6 self)
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We prove an analog of Böttcher’s theorem for transcendental entire functions in the EremenkoLyubich class B. More precisely, let f and g be entire functions with bounded sets of singular values and suppose that f and g belong to the same parameter space (i.e., are quasiconformally equivalent in the sense of Eremenko and Lyubich). Then f and g are conjugate when restricted to the set of points which remain in some sufficiently small neighborhood of infinity under iteration. Furthermore, this conjugacy extends to a quasiconformal selfmap of the plane. We also prove that this conjugacy is essentially unique. In particular, we show that a function f ∈ B has no invariant line fields on its escaping set. Finally, we show that any two hyperbolic functions f, g ∈ B which belong to the same parameter space are conjugate on their sets of escaping points.
Comparison Theorems And Orbit Counting In Hyperbolic Geometry
 Trans. Amer. Math. Soc
, 1998
"... In this article we address an interesting problem in hyperbolic geometry. This is the problem of comparing different quantities associated to the fundamental group of a hyperbolic manifold (e.g. word length, displacement in the universal cover, etc.) asymptotically. Our method involves a mixture of ..."
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Cited by 13 (11 self)
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In this article we address an interesting problem in hyperbolic geometry. This is the problem of comparing different quantities associated to the fundamental group of a hyperbolic manifold (e.g. word length, displacement in the universal cover, etc.) asymptotically. Our method involves a mixture of ideas from both "thermodynamic" ergodic theory and the automaton associated to strongly Markov groups. 0.
Approximation by maximal cusps in boundaries of deformation spaces of Kleinian groups
 J. Differential Geom
"... Let M be a compact, oriented, irreducible, atoroidal 3manifold with nonempty boundary. Let CC0(M) denote the space of convex cocompact Kleinian groups uniformizing M. We show that any Kleinian group in the boundary of CC0(M) whose limit set is the whole sphere can be approximated by maximal cusps. ..."
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Cited by 11 (1 self)
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Let M be a compact, oriented, irreducible, atoroidal 3manifold with nonempty boundary. Let CC0(M) denote the space of convex cocompact Kleinian groups uniformizing M. We show that any Kleinian group in the boundary of CC0(M) whose limit set is the whole sphere can be approximated by maximal cusps. Density of maximal cusps on the boundary of Schottky space is derived as a corollary. We further show that maximal cusps are dense in the boundary of the quasiconformal deformation space of any geometrically finite hyperbolic 3manifold with connected conformal boundary. 1.
Wandering domains and invariant conformal structures for mappings of the 2torus
"... A C 1 circle diffeomorphism with irrational rotation number need not have any dense orbits. However, any C² circle diffeomorphism with irrational rotation number must in fact be topologically conjugate to an irrational rotation. This paper addresses the analogous matter for the 2torus. We say tha ..."
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Cited by 9 (3 self)
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A C 1 circle diffeomorphism with irrational rotation number need not have any dense orbits. However, any C² circle diffeomorphism with irrational rotation number must in fact be topologically conjugate to an irrational rotation. This paper addresses the analogous matter for the 2torus. We say that a diffeomorphism f of T², isotopic to the identity, has Denjoy type if hf = Rh, where R is some minimal translation of the torus, and h is a continuous torus mapping homotopic to the identity such that {x ∈ T²: cardinality(h −1 (x))> 1} is nonempty and countable. If f has Denjoy type, the interior of any fiber h −1 (x), if nonempty, is a wandering domain for f. It is known that there are C² diffeomorphisms of Denjoy type, but not known whether they can be C³. Our main results imply the following Theorem. Let f ∈ Diff 1 (T²) have Denjoy type, with minimal set Γ = T². (i) If f preserves a measurable, essentially bounded conformal structure on Γ, then the collection {f n} (considered as mappings of the ideal boundaries of the wandering domains) has unbounded quasisymmetric distortion, and