2,179 research outputs found
Flats in spaces with convex geodesic bicombings
In spaces of nonpositive curvature the existence of isometrically embedded
flat (hyper)planes is often granted by apparently weaker conditions on large
scales. We show that some such results remain valid for metric spaces with
non-unique geodesic segments under suitable convexity assumptions on the
distance function along distinguished geodesics. The discussion includes, among
other things, the Flat Torus Theorem and Gromov's hyperbolicity criterion
referring to embedded planes. This generalizes results of Bowditch for Busemann
spaces.Comment: Final version, to appear in Analysis and Geometry in Metric Spaces
(AGMS
Asymptotic behavior of averaged and firmly nonexpansive mappings in geodesic spaces
We further study averaged and firmly nonexpansive mappings in the setting of
geodesic spaces with a main focus on the asymptotic behavior of their Picard
iterates. We use methods of proof mining to obtain an explicit quantitative
version of a generalization to geodesic spaces of result on the asymptotic
behavior of Picard iterates for firmly nonexpansive mappings proved by Reich
and Shafrir. From this result we obtain effective uniform bounds on the
asymptotic regularity for firmly nonexpansive mappings. Besides this, we derive
effective rates of asymptotic regularity for sequences generated by two
algorithms used in the study of the convex feasibility problem in a nonlinear
setting
Holomorphic motions of weighted periodic points
We study the holomorphic motions of repelling periodic points in stable
families of endomorphisms of . In particular, we
establish an asymptotic equidistribution of the graphs associated to such
periodic points with respect to natural measures in the space of all
holomorphic motions of points in the Julia sets
Dold sequences, periodic points, and dynamics
In this survey we describe how the so-called Dold congruence arises in
topology, and how it relates to periodic point counting in dynamical systems.Comment: 38 pages; surve
Expansiveness, Lyapunov exponents and entropy for set valued maps
In this paper we introduce a notion of expansiveness for a set valued map
defined on a topological space different from that given by Richard Williams at
\cite{Wi, Wi2} and prove that the topological entropy of an expansive set
valued map defined on a Peano space of positive dimension is greater than zero.
We define Lyapunov exponent for set valued maps and prove that positiveness of
its Lyapunov exponent implies positiveness for the topological entropy. Finally
we introduce the definition of (Lyapunov) stable points for set valued maps and
prove a dichotomy for the set of stable points for set valued maps defined on
Peano spaces: either it is empty or the whole space.Comment: 24 pages, 1 figur
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
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