20 research outputs found
Cyclic projectors and separation theorems in idempotent convex geometry
Semimodules over idempotent semirings like the max-plus or tropical semiring
have much in common with convex cones. This analogy is particularly apparent in
the case of subsemimodules of the n-fold cartesian product of the max-plus
semiring it is known that one can separate a vector from a closed subsemimodule
that does not contain it. We establish here a more general separation theorem,
which applies to any finite collection of closed semimodules with a trivial
intersection. In order to prove this theorem, we investigate the spectral
properties of certain nonlinear operators called here idempotent cyclic
projectors. These are idempotent analogues of the cyclic nearest-point
projections known in convex analysis. The spectrum of idempotent cyclic
projectors is characterized in terms of a suitable extension of Hilbert's
projective metric. We deduce as a corollary of our main results the idempotent
analogue of Helly's theorem.Comment: 20 pages, 1 figur
Minimal half-spaces and external representation of tropical polyhedra
We give a characterization of the minimal tropical half-spaces containing a
given tropical polyhedron, from which we derive a counter example showing that
the number of such minimal half-spaces can be infinite, contradicting some
statements which appeared in the tropical literature, and disproving a
conjecture of F. Block and J. Yu. We also establish an analogue of the
Minkowski-Weyl theorem, showing that a tropical polyhedron can be equivalently
represented internally (in terms of extreme points and rays) or externally (in
terms of half-spaces containing it). A canonical external representation of a
polyhedron turns out to be provided by the extreme elements of its tropical
polar. We characterize these extreme elements, showing in particular that they
are determined by support vectors.Comment: 19 pages, 4 figures, example added with a new figure, figures
improved, references update
Computing the vertices of tropical polyhedra using directed hypergraphs
We establish a characterization of the vertices of a tropical polyhedron
defined as the intersection of finitely many half-spaces. We show that a point
is a vertex if, and only if, a directed hypergraph, constructed from the
subdifferentials of the active constraints at this point, admits a unique
strongly connected component that is maximal with respect to the reachability
relation (all the other strongly connected components have access to it). This
property can be checked in almost linear-time. This allows us to develop a
tropical analogue of the classical double description method, which computes a
minimal internal representation (in terms of vertices) of a polyhedron defined
externally (by half-spaces or hyperplanes). We provide theoretical worst case
complexity bounds and report extensive experimental tests performed using the
library TPLib, showing that this method outperforms the other existing
approaches.Comment: 29 pages (A4), 10 figures, 1 table; v2: Improved algorithm in section
5 (using directed hypergraphs), detailed appendix; v3: major revision of the
article (adding tropical hyperplanes, alternative method by arrangements,
etc); v4: minor revisio
Stable transitivity of certain noncompact extensions of hyperbolic systems
Let ƒ : X → X be the restriction to a hyperbolic basic set of a smooth diffeomorphism. We find several criteria for transitivity of noncompact connected Lie group extensions. As a consequence, we find transitive extensions for any finite-dimensional connected Lie group extension. If, in addition, the group is perfect and has an open set of elements that generate a compact subgroup, we find open sets of stably transitive extensions. In particular, we find stably transitive SL(2, R)-extensions. More generally, we find stably transitive Sp(2n-R)-extensions for all n ≥ 1. For the Euclidean groups SE(n) with n ≥ 4 even, we obtain a new proof of a result of Melbourne and Nicol stating that there is an open and dense set of extensions that are transitive. For groups of the form K × Rn where K is compact, a separation condition is necessary for transitivity. Provided X is a hyperbolic attractor, we show that an open and dense set of extensions satisfying the separation condition are transitive. This generalises a result of Nitica and Pollicott for Rn-extensions.</p
Transitivity of Euclidean-type extensions of hyperbolic systems
Let f:X→X be the restriction to a hyperbolic basic set of a smooth diffeomorphism. We show that in the class of Cr(r>0) cocycles with fiber the special Euclidean group SE(n), those that are transitive form a residual set (countable intersection of open dense sets). This result is new for odd values of n≥3. More generally, we consider Euclidean-type groups Gn where G is a compact connected Lie group acting linearly on n. When Fix G={0}, it is again the case that the transitive cocycles are residual. When Fix G≠{0}, the same result holds upon restriction to the subset of cocycles that avoid an obvious and explicit obstruction to transitivity