1,461 research outputs found
Conjugacies of model sets
Let be a model set meeting two simple conditions: (1) the internal space
is a product of and a finite group, and (2) the window is a
finite union of disjoint polyhedra. Then any point pattern with finite local
complexity (FLC) that is topologically conjugate to is mutually locally
derivable (MLD) to a model set that has the same internal group and window
as , but has a different projection from to . In
cohomological terms, this means that the group of
asymptotically negligible classes has dimension . We also exhibit a
counterexample when the second hypothesis is removed, constructing two
topologically conjugate FLC Delone sets, one a model set and the other not even
a Meyer set.Comment: Updated to the published versio
From local to global analytic conjugacies
Let and be rational maps with Julia sets and , and let be any continuous map such that on . We show that if is -differentiable, with non-vanishing derivative, at some repelling periodic point , then admits an analytic extension to , where is the exceptional set of . Moreover, this extension is a semiconjugacy. This generalizes a result of Julia (Ann. Sci. École Norm. Sup. (3) 40 (1923), 97–150). Furthermore, if then the extended map is rational, and in this situation and , provided that is not constant. On the other hand, if then the extended map may be transcendental: for example, when is a power map (conjugate to ) or a Chebyshev map (conjugate to \pm \text{Х}_d with \text{Х}_d(z+z^{-1}) = z^d+z^{-d}), and when is an integral Lattès example (a quotient of the multiplication by an integer on a torus). Eremenko (Algebra i Analiz 1(4) (1989), 102–116) proved that these are the only such examples. We present a new proof
Explosion of smoothness for conjugacies between multimodal maps
Let and be smooth multimodal maps with no periodic attractors and no
neutral points. If a topological conjugacy between and is
at a point in the nearby expanding set of , then is a smooth
diffeomorphism in the basin of attraction of a renormalization interval of .
In particular, if and are unimodal maps and
is at a boundary of then is in .Comment: 22 page
Linearizability of Saturated Polynomials
Brjuno and R\"ussmann proved that every irrationally indifferent fixed point
of an analytic function with a Brjuno rotation number is linearizable, and
Yoccoz proved that this is sharp for quadratic polynomials. Douady conjectured
that this is sharp for all rational functions of degree at least 2, i.e., that
non-M\"obius rational functions cannot have Siegel disks with non-Brjuno
rotation numbers. We prove that Douady's conjecture holds for the class of
polynomials for which the number of infinite tails of critical orbits in the
Julia set equals the number of irrationally indifferent cycles. As a corollary,
Douady's conjecture holds for the polynomials for all
and all complex .Comment: 28 pages, major revisions and additions following referee comment
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