- journal articles metadata
European Mathematical Society Publishing House
2024-03-29 13:07:35
7
https://www.ems-ph.org/meta/jmeta-stream.php?jrn=CMH&vol=75&iss=4&update_since=2024-03-29
Commentarii Mathematici Helvetici
Comment. Math. Helv.
CMH
0010-2571
1420-8946
General
10.4171/CMH
http://www.ems-ph.org/doi/10.4171/CMH
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European Mathematical Society Publishing House
Zuerich, Switzerland
© Swiss Mathematical Society
75
2000
4
Hausdorff dimension and conformal dynamics II: Geometrically finite rational maps
Curtis
McMullen
Harvard University, CAMBRIDGE, UNITED STATES
Complex dynamics, iterated rational maps, Julia sets, Hausdorff dimension
This paper investigates several dynamically defined dimensions for rational maps f on the Riemann sphere, providing a systematic treatment modeled on the theory for Kleinian groups. We begin by defining the radial Julia set Jrad(f), and showing that every rational map satisfies $ {\rm H.\,dim}\,J_{{\rm rad}}(f) = \alpha(f) $ where $ \alpha(f) $ is the minimal dimension of an f-invariant conformal density on the sphere. A rational map f is geometrically finite if every critical point in the Julia set is preperiodic. In this case we show {\rm H.\,dim}\,J_{{\rm rad}}(f) = {\rm H.\,dim}\,J(f) = \delta (f) $, where $ \delta(f) $ is the critical exponent of the Poincaré series; and f admits a unique normalized invariant density 7 of dimension $ \delta(f) $. Now let f be geometrically finite and suppose $ f_n \to f $ algebraically, preserving critical relations. When the convergence is horocyclic for each parabolic point of f, we show fn is geometrically finite for $ n \gg 0 $ and $ J(f_n) \to J(f) $ in the Hausdorff topology. If the convergence is radial, then in addition we show $ {\rm H.\,dim}\,J(f_{n}) \to {\rm H.\,dim}\,J(f) $. We give examples of horocyclic but not radial convergence where $ {\rm H.\,dim}\,J(f_{n}) \to 1 > {\rm H.\,dim}\,J(f) = 1/2 + \epsilon $. We also give a simple demonstration of Shishikura's result that there exist $ f_n(z) = z^2 + c_n $ with $ {\rm H.\,dim}\,J(f_{n}) \to 2 $. The proofs employ a new method that reduces the study of parabolic points to the case of elementary Kleinian groups.
Dynamical systems and ergodic theory
General
535
593
10.1007/s000140050140
http://www.ems-ph.org/doi/10.1007/s000140050140
Asymptotic degeneration of representations of quivers
V.
Strassen
Universität Konstanz, KONSTANZ, GERMANY
Quiver, representation, degeneration, tensor product, asymptotics
We define asymptotic degeneration of nilpotent representations of an arbitrary finite quiver, using large tensor powers and small direct sums, and characterize this notion by a simple and effective criterion.
Associative rings and algebras
General
594
607
10.1007/s000140050141
http://www.ems-ph.org/doi/10.1007/s000140050141
Strong approximation for Zariski dense subgroups over arbitrary global fields
Richard
Pink
ETH Zürich, ZÜRICH, SWITZERLAND
Consider a finitely generated Zariski dense subgroup $ \Gamma $ of a connected simple algebraic group G over a global field F. An important aspect of strong approximation is the question of whether the closure of $ \Gamma $ in the group of points of G with coefficients in a ring of partial adeles is open. We prove an essentially optimal result in this direction, based on the condition that $ \Gamma $ is not discrete in that ambient group. There are no restrictions on the characteristic of F or the type of G, and simultaneous approximation in finitely many algebraic groups is also studied. Classification of finite simple groups is not used.
Group theory and generalizations
General
608
643
10.1007/s000140050142
http://www.ems-ph.org/doi/10.1007/s000140050142
Geometry for palindromic automorphism groups of free groups
Henry
Glover
Ohio State University, COLUMBUS, UNITED STATES
C.
Jensen
University of New Orleans, NEW ORLEANS, UNITED STATES
Group theory and generalizations
General
644
667
10.1007/s000140050143
http://www.ems-ph.org/doi/10.1007/s000140050143
Weierstrass representation of Lagrangian surfaces in four-dimensional space using spinors and quaternions
F.
Hélein
CMLA-ENS de Cachan, CACHAN CEDEX, FRANCE
P.
Romon
Université de Marne la Vallée, MARNE LA VALLÉE CEDEX 2, FRANCE
Manifolds and cell complexes
General
668
680
10.1007/s000140050144
http://www.ems-ph.org/doi/10.1007/s000140050144
Symplectic invariants of elliptic fixed points
K.
Siburg
Ruhr-Universität Bochum, BOCHUM, GERMANY
Elliptic fixed point, Birkhoff normal form, Aubry--Mather theory, minimal action, Reeb flow, period spectrum, geodesic flow, length spectrum
To the germ of an area--preserving diffeomorphism at an elliptic fixed point, we associate the germ of Mather's minimal action. This yields a strictly convex function which is symplectically invariant and comprises the classical Birkhoff invariants as the Taylor coefficients of its convex conjugate. In addition, however, the minimal action contains information about the local dynamics near the fixed point; for instance, it detects the C0--integrability of the diffeomorphism. Applied to the Reeb flow, this leads to new period spectrum invariants for three--dimensional contact manifolds; a particular case is the geodesic flow on a two--dimensional Riemannian manifold, where the period spectrum is the classical length spectrum.
Global analysis, analysis on manifolds
General
681
700
10.1007/s000140050145
http://www.ems-ph.org/doi/10.1007/s000140050145
Rigidity of quasi-isometries for some hyperbolic buildings
Marc
Bourdon
Université Lille I, VILLENEUVE D'ASCQ CEDEX, FRANCE
H.
Pajot
Université de Cergy-Pontoise, CERGY-PONTOISE CEDEX, FRANCE
Geometry
General
701
736
10.1007/s000140050146
http://www.ems-ph.org/doi/10.1007/s000140050146