5,105 research outputs found
Preservation of External Rays in non-Autonomous Iteration
We consider the dynamics arising from the iteration of an arbitrary sequence
of polynomials with uniformly bounded degrees and coefficients and show that,
as parameters vary within a single hyperbolic component in parameter space,
certain properties of the corresponding Julia sets are preserved. In
particular, we show that if the sequence is hyperbolic and all the Julia sets
are connected, then the whole basin at infinity moves holomorphically. This
extends also to the landing points of external rays and the resultant
holomorphic motion of the Julia sets coincides with that obtained earlier using
grand orbits. In addition, if a finite set of external rays separate the Julia
set for a particular parameter value, then the rays with the same external
angles separate the Julia set for every parameter in the same hyperbolic
component
Distorted plane waves on manifolds of nonpositive curvature
We will consider the high frequency behaviour of distorted plane waves on
manifolds of nonpositive curvature which are Euclidean or hyperbolic near
infinity, under the assumption that the curvature is negative close to the
trapped set of the geodesic flow and that the topological pressure associated
to half the unstable Jacobian is negative.
We obtain a precise expression for distorted plane waves in the high
frequency limit, similar to the one in \cite{GN} in the case of convex
co-compact manifolds. In particular, we will show bounds on
distorted plane waves that are uniform with frequency. We will also show that
the real part of distorted plane waves restricted to a compact set satisfy the
analogue of Yau's conjecture about the Haussdorff measure of nodal sets.Comment: 48 pages, 5 figures. A few corrections and new results (concerning
small-scale equidistribution) were adde
Regularity at infinity of real mappings and a Morse-Sard theorem
We prove a new Morse-Sard type theorem for the asymptotic critical values of
semi-algebraic mappings and a new fibration theorem at infinity for
mappings. We show the equivalence of three different types of regularity
conditions which have been used in the literature in order to control the
asymptotic behaviour of mappings. The central role of our picture is played by
the -regularity and its bridge toward the -regularity which implies
topological triviality at infinity
Generalized polygons with non-discrete valuation defined by two-dimensional affine R-buildings
In this paper, we show that the building at infinity of a two-dimensional
affine R-building is a generalized polygon endowed with a valuation satisfying
some specific axioms. Specializing to the discrete case of affine buildings,
this solves part of a long standing conjecture about affine buildings of type
G~_2, and it reproves the results obtained mainly by the second author for
types A~_2 and C~_2. The techniques are completely different from the ones
employed in the discrete case, but they are considerably shorter, and general
(i.e., independent of the type of the two-dimensional R-building)
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