874 research outputs found
On the "Mandelbrot set" for a pair of linear maps and complex Bernoulli convolutions
We consider the "Mandelbrot set" for pairs of complex linear maps,
introduced by Barnsley and Harrington in 1985 and studied by Bousch, Bandt and
others. It is defined as the set of parameters in the unit disk such
that the attractor of the IFS is
connected. We show that a non-trivial portion of near the imaginary axis is
contained in the closure of its interior (it is conjectured that all non-real
points of are in the closure of the set of interior points of ). Next we
turn to the attractors themselves and to natural measures
supported on them. These measures are the complex analogs of
much-studied infinite Bernoulli convolutions. Extending the results of Erd\"os
and Garsia, we demonstrate how certain classes of complex algebraic integers
give rise to singular and absolutely continuous measures . Next we
investigate the Hausdorff dimension and measure of , for
in the set , for Lebesgue-a.e. . We also obtain partial results on
the absolute continuity of for a.e. of modulus greater
than .Comment: 22 pages, 5 figure
Salem numbers and Pisot numbers via interlacing
We present a general construction of Salem numbers via rational functions
whose zeros and poles mostly lie on the unit circle and satisfy an interlacing
condition. This extends and unifies earlier work. We then consider the
"obvious" limit points of the set of Salem numbers produced by our theorems,
and show that these are all Pisot numbers, in support of a conjecture of Boyd.
We then show that all Pisot numbers arise in this way. Combining this with a
theorem of Boyd, we show that all Salem numbers are produced via an interlacing
construction.Comment: 21 pages, 5 figures, updated in response to reviewer comment
The dynamical Manin-Mumford problem for plane polynomial automorphisms
Let be a polynomial automorphism of the affine plane. In this paper we
consider the possibility for it to possess infinitely many periodic points on
an algebraic curve . We conjecture that this happens if and only if
admits a time-reversal symmetry; in particular the Jacobian
must be a root of unity.
As a step towards this conjecture, we prove that the Jacobian of and all
its Galois conjugates lie on the unit circle in the complex plane. Under mild
additional assumptions we are able to conclude that indeed is
a root of unity. We use these results to show in various cases that any two
automorphisms sharing an infinite set of periodic points must have a common
iterate, in the spirit of recent results by Baker-DeMarco and Yuan-Zhang.Comment: 45 pages. Theorems A and B are now extended to automorphisms defined
over any field of characteristic zer
Shift Radix Systems - A Survey
Let be an integer and . The {\em shift radix system} is defined by has the {\em finiteness
property} if each is eventually mapped to
under iterations of . In the present survey we summarize
results on these nearly linear mappings. We discuss how these mappings are
related to well-known numeration systems, to rotations with round-offs, and to
a conjecture on periodic expansions w.r.t.\ Salem numbers. Moreover, we review
the behavior of the orbits of points under iterations of with
special emphasis on ultimately periodic orbits and on the finiteness property.
We also describe a geometric theory related to shift radix systems.Comment: 45 pages, 16 figure
Distortion in transformation groups
We exhibit rigid rotations of spheres as distortion elements in groups of diffeomorphisms, thereby answering a question of J Franks and M Handel. We also show that every homeomorphism of a sphere is, in a suitable sense, as distorted as possible in the group Homeo(Sn), thought of as a discrete group.
An appendix by Y de Cornulier shows that Homeo(Sn) has the strong boundedness property, recently introduced by G Bergman. This means that every action of the discrete group Homeo(Sn) on a metric space by isometries has bounded orbits
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