4,330 research outputs found
Connected Choice and the Brouwer Fixed Point Theorem
We study the computational content of the Brouwer Fixed Point Theorem in the
Weihrauch lattice. Connected choice is the operation that finds a point in a
non-empty connected closed set given by negative information. One of our main
results is that for any fixed dimension the Brouwer Fixed Point Theorem of that
dimension is computably equivalent to connected choice of the Euclidean unit
cube of the same dimension. Another main result is that connected choice is
complete for dimension greater than or equal to two in the sense that it is
computably equivalent to Weak K\H{o}nig's Lemma. While we can present two
independent proofs for dimension three and upwards that are either based on a
simple geometric construction or a combinatorial argument, the proof for
dimension two is based on a more involved inverse limit construction. The
connected choice operation in dimension one is known to be equivalent to the
Intermediate Value Theorem; we prove that this problem is not idempotent in
contrast to the case of dimension two and upwards. We also prove that Lipschitz
continuity with Lipschitz constants strictly larger than one does not simplify
finding fixed points. Finally, we prove that finding a connectedness component
of a closed subset of the Euclidean unit cube of any dimension greater or equal
to one is equivalent to Weak K\H{o}nig's Lemma. In order to describe these
results, we introduce a representation of closed subsets of the unit cube by
trees of rational complexes.Comment: 36 page
An index for Brouwer homeomorphisms and homotopy Brouwer theory
We use the homotopy Brouwer theory of Handel to define a Poincar{\'e} index
between two orbits for an orientation preserving fixed point free homeomorphism
of the plane. Furthermore, we prove that this index is almost additive.Comment: To appear in Ergodic Theory and Dynamical System
Strictly Toral Dynamics
This article deals with nonwandering (e.g. area-preserving) homeomorphisms of
the torus which are homotopic to the identity and strictly
toral, in the sense that they exhibit dynamical properties that are not present
in homeomorphisms of the annulus or the plane. This includes all homeomorphisms
which have a rotation set with nonempty interior. We define two types of
points: inessential and essential. The set of inessential points is
shown to be a disjoint union of periodic topological disks ("elliptic
islands"), while the set of essential points is an essential
continuum, with typically rich dynamics (the "chaotic region"). This
generalizes and improves a similar description by J\"ager. The key result is
boundedness of these "elliptic islands", which allows, among other things, to
obtain sharp (uniform) bounds of the diffusion rates. We also show that the
dynamics in is as rich as in from the rotational
viewpoint, and we obtain results relating the existence of large invariant
topological disks to the abundance of fixed points.Comment: Incorporates suggestions and corrections by the referees. To appear
in Inv. Mat
The Complexity of the Homotopy Method, Equilibrium Selection, and Lemke-Howson Solutions
We show that the widely used homotopy method for solving fixpoint problems,
as well as the Harsanyi-Selten equilibrium selection process for games, are
PSPACE-complete to implement. Extending our result for the Harsanyi-Selten
process, we show that several other homotopy-based algorithms for finding
equilibria of games are also PSPACE-complete to implement. A further
application of our techniques yields the result that it is PSPACE-complete to
compute any of the equilibria that could be found via the classical
Lemke-Howson algorithm, a complexity-theoretic strengthening of the result in
[Savani and von Stengel]. These results show that our techniques can be widely
applied and suggest that the PSPACE-completeness of implementing homotopy
methods is a general principle.Comment: 23 pages, 1 figure; to appear in FOCS 2011 conferenc
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