Almost periodically forced circle flows

We study general dynamical and topological behaviors of minimal sets in skew-product circle flows in both continuous and discrete settings, with particular attentions paying to almost periodically forced circle flows. When a circle flow is either discrete in time and unforced (i.e., a circle map) or continuous in time but periodically forced, behaviors of minimal sets are completely characterized by classical theory. The general case involving almost periodic forcing is much more complicated due to the presence of multiple forcing frequencies, the topological complexity of the forcing space, and the possible loss of mean motion property. On one hand, we will show that to some extent behaviors of minimal sets in an almost periodically forced circle flow resemble those of Denjoy sets of circle maps in the sense that they can be almost automorphic, Cantorian, and everywhere non-locally connected. But on the other hand, we will show that almost periodic forcing can lead to significant topological and dynamical complexities on minimal sets which exceed the contents of Denjoy theory. For instance, an almost periodically forced circle flow can be positively transitive and its minimal sets can be Li-Yorke chaotic and non-almost automorphic. As an application of our results, we will give a complete classification of minimal sets for the projective bundle flow of an almost periodic, sl(2,R)-valued, continuous or discrete cocycle.Continuous almost periodically forced circle flows are among the simplest non-monotone, multi-frequency dynamical systems. They can be generated from almost periodically forced nonlinear oscillators through integral manifolds reduction in the damped cases and through Mather theory in the damping-free cases. They also naturally arise in 2D almost periodic Floquet theory as well as in climate models. Discrete almost periodically forced circle flows arise in the discretization of nonlinear oscillators and discrete counterparts of linear Schrödinger equations with almost periodic potentials. They have been widely used as models for studying strange, non-chaotic attractors and intermittency phenomena during the transition from order to chaos. Hence the study of these flows is of fundamental importance to the understanding of multi-frequency-driven dynamical irregularities and complexities in non-monotone dynamical systems.     

Composition formulas in the Weyl calculus

In pseudodifferential analysis, the usual composition formula, which has asymptotic value, extends that valid for differential operators. The one developed here is based instead on the decomposition of symbols (functions in Rn×Rn) as integral superpositions of homogeneous ones, of degrees lying on the complex line with real part −n. It extends the one known in the one-dimensional case in connection with automorphic pseudodifferential analysis.     

A Note on non-autonomous scalar functional differential equations with small delay

We prove that the minimal sets in the skew-product semiflows generated from a non-autonomous scalar functional differential equation with a small delay are all almost automorphic extensions of the base. This result is not true for arbitrary delay equations. The point is that, for a small delay, so-called special solutions exist and permit us to tackle the problem by means of some related scalar ODE's for which the study is much simpler. To cite this article: A.I. Alonso et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

On Shalika models and <Emphasis Type="Italic">p</Emphasis>-adic <Emphasis Type="Italic">L</Emphasis>-functions

We use modular symbols to construct p-adic L-functions for cohomological cuspidal automorphic representations on GL(2n), which admit a Shalika model. Our construction differs from former ones in that it systematically makes use of the representation theory of p-adic groups.     

Almost automorphic symbolic minimal sets without unique ergodicity

Given a metrizable monothetic groupG with generatorg and a suitable closed nowhere dense subsetC of positive Haar measure, we associate a natural compact metric space whose points are almost automorphic symbolic minimal sets. It is then shown that those minimal sets which have positive topological entropy and fail to be uniquely ergodic form a esidual set. The example due to P. Julius [2] of a Toeplitz sequence of positive entropy which, is uniquely ergodic shows that the “residual” conclusion is sharp.     

Finiteness and recognizability problems for substitution maps on two symbols

We investigate minimal, compact, square-closed subsets of the unit circle by identifying them with subsets of [0,1] and with sets of infinite words on two symbols; in particular, such sets arising from substitution maps. We consider problems on finiteness, recognizability, square roots, density and measure. Properties of the free semigroup on two symbols play a significant role in the analysis.     

A construction of almost automorphic minimal sets

We describe a general procedure to construct topological extensions of given skew product maps with one-dimensional fibres by blowing up a countable number of single points to vertical segments. This allows to produce various examples of unusual dynamics, including almost automorphic minimal sets of almost periodically forced systems, point-distal but non-distal homeomorphisms of the torus (as first constructed by Rees) or minimal sets of quasiperiodically forced interval maps which are not filled-in.     

Combinatorics on partial word correlations

Partial words are strings over a finite alphabet that may contain a number of “do not know” symbols. In this paper, we consider the period and weak period sets of partial words of length n over a finite alphabet, and study the combinatorics of specific representations of them, called correlations, which are binary and ternary vectors of length n indicating the periods and weak periods. We characterize precisely which vectors represent the period and weak period sets of partial words and prove that all valid correlations may be taken over the binary alphabet. We show that the sets of all such vectors of a given length form distributive lattices under suitably defined partial orderings. We show that there is a well-defined minimal set of generators for any binary correlation of length n and demonstrate that these generating sets are the primitive subsets of {1,2,…,n−1}. We also investigate the number of partial word correlations of length n. Finally, we compute the population size, that is, the number of partial words sharing a given correlation, and obtain recurrences to compute it. Our results generalize those of Guibas, Odlyzko, Rivals and Rahmann.     

Almost automorphic solutions for some evolution equations through the minimizing for some subvariant functional, applications to heat and wave equations with nonlinearities

In this work, we study the existence of bounded and almost automorphic solutions for evolution equations in Banach spaces. We suppose that the linear part is the infinitesimal generator of a compact C₀-semigroup of bounded linear operators and the nonlinear part is an almost automorphic function with respect to the second argument. We give sufficient conditions ensuring the existence of an almost automorphic solution when there is at least one bounded solution on R⁺. We use the subvariant functional method to show that every K-minimizing mild solution is compact almost automorphic. Applications are provided for both heat and wave equations with nonlinearities in several functional spaces.     

Almost automorphic solutions to nonautonomous semilinear evolution equations in Banach spaces

This paper is concerned with almost automorphy of the solutions to a nonautonomous semilinear evolution equation u^′(t)=A(t)u(t)+f(t,u(t)) in a Banach space with a Stepanov-like almost automorphic nonlinear term. We establish a composition theorem for Stepanov-like almost automorphic functions. Furthermore, we obtain some existence and uniqueness theorems for almost automorphic solutions to the nonautonomous evolution equation, by means of the evolution family and the exponential dichotomy. Some results in this paper are new even if A(t) is time independent.     

Small point sets of PG(n, q 3) intersecting each k-subspace in 1 mod q points

The main result of this paper is that point sets of PG(n, q ³), q = p ^h , p ≥ 7 prime, of size less than 3(q ^3(n?k) + 1)/2 intersecting each k-space in 1 modulo q points (these are always small minimal blocking sets with respect to k-spaces) are linear blocking sets. As a consequence, we get that minimal blocking sets of PG(n, p ³), p ≥ 7 prime, of size less than 3(p ^3(n?k) + 1)/2 with respect to k-spaces are linear. We also give a classification of small linear blocking sets of PG(n, q ³) which meet every (n ? 2)-space in 1 modulo q points.     

Characterization results on small blocking sets of the polar spaces Q +(2n + 1, 2) and Q +(2n + 1, 3)

In De Beule and Storme, Des Codes Cryptogr 39(3):323–333, De Beule and Storme characterized the smallest blocking sets of the hyperbolic quadrics Q ⁺(2n + 1, 3), n ≥ 4; they proved that these blocking sets are truncated cones over the unique ovoid of Q ⁺(7, 3). We continue this research by classifying all the minimal blocking sets of the hyperbolic quadrics Q ⁺(2n + 1, 3), n ≥ 3, of size at most 3 ⁿ + 3 ^n–2. This means that the three smallest minimal blocking sets of Q ⁺(2n + 1, 3), n ≥ 3, are now classified. We present similar results for q = 2 by classifying the minimal blocking sets of Q ⁺(2n + 1, 2), n ≥ 3, of size at most 2 ⁿ + 2 ^n-2. This means that the two smallest minimal blocking sets of Q ⁺(2n + 1, 2), n ≥ 3, are classified.     

Levels in the toposes of simplicial sets and cubical sets

The essential subtoposes of a fixed topos form a complete lattice, which gives rise to the notion of a level in a topos. In the familiar example of simplicial sets, levels coincide with dimensions and give rise to the usual notions of n-skeletal and n-coskeletal simplicial sets. In addition to the obvious ordering, the levels provide a stricter means of comparing the complexity of objects, which is determined by the answer to the following question posed by Bill Lawvere: when does n-skeletal imply k-coskeletal? This paper, which subsumes earlier unpublished work of some of the authors, answers this question for several toposes of interest to homotopy theory and higher category theory: simplicial sets, cubical sets, and reflexive globular sets. For the latter, n-skeletal implies (n+1)-coskeletal but for the other two examples the situation is considerably more complicated: n-skeletal implies (2n−1)-coskeletal for simplicial sets and 2n-coskeletal for cubical sets, but nothing stronger. In a discussion of further applications, we prove that n-skeletal cyclic sets are necessarily (2n+1)-coskeletal.     

Chebyshev polynomials for disjoint compact sets

We are concerned with the problem of finding among all polynomials of degreen with leading coefficient 1, the one which has minimal uniform norm on the union of two disjoint compact sets in the complex plane. Our main object here is to present a class of disjoint sets where the best approximation can be determined explicitly for alln. A closely related approximation problem is obtained by considering all polynomials that have degree no larger thann and satisfy an interpolatory constraint. Such problems arise in certain iterative matrix computations. Again, we discuss a class of disjoint compact sets where the optimal polynomial is explicitly known for alln.Communicated by Doron S. Lubinsky     

Edge-Ramsey theory

LetK be a configuration, a set of points in some finite-dimensional Euclidean space. Letn andk be positive integers. The notationR(K, n, r) is an abbreviation for the following statement: For everyr-coloring of the points of then-dimensional Euclidean space,R ⁿ , a monochromatic configurationL which is congruent toK exists. A configurationK is Ramsey if the following holds: For every positive integerr, a positive integern=n(K, r) exists such that, for allm≥n, R(K, m, r) holds. A configuration is spherical if it can be embedded in the surface of a sphere inn-space, providedn is sufficiently large. It is relatively easy to show that if a configuration is Ramsey, it must be spherical. Accordingly, a good fraction of the research efforts in Euclidean Ramsey theory is devoted to determining which spherical configurations are Ramsey. It is known that then-dimensional measure polytopes (the higher-dimensional analogs of a cube), then-dimensional simplex, and the regular polyhedra inR ² andR ³ are Ramsey. Now letE denote a set of edges in a configurationK. The pair (K, E) is called an edge-configuration, andR _e (K, E, n, r) is used as an abbreviation for the following statement: For anyr-coloring of the edges ofR ⁿ , there is an edge configuration (L, F) congruent to (K, E) so that all edges inF are assigned the same color. An edge-configuration isedge-Ramsey if, for allr≥1, a positive integern=n(K, E, r) exists so that ifm≥n, the statementR _e (K, E, m, r) holds. IfK is a regular polytope, it is saidK isedge-Ramsey when the configuration determined by the set of edges of minimum length is edge-Ramsey. It is known that then-dimensional simplex is edge-Ramsey and that the nodes of any edge-Ramsey configuration can be partitioned into two spherical sets. Furthermore, the edges of any edge-Ramsey configuration must all have the same length. It is conjectured that the unit square is edge-Ramsey, and it is natural to ask the more general question: Which regular polytopes are edge-Ramsey? In this article it is shown that then-dimensional measure polytope and then-dimensional cross polytope are edge-Ramsey. It is also shown that these two infinite families and then-dimensional simplexes are the only regular edge-Ramsey polytopes, with the possible exceptions of the hexagon and the 24-cell.     

Some topological properties of weakly almost periodic mappings

A map φ:X→X induces a linear operator T:C(X)→C(X) by composition: Tf(x)=f°φ(x). T and φ are termed weakly almost periodic if the sequence {Tⁿ} is precompact in the weak operator topology. Using general structure theorems for weakly almost periodic operators, the properties of these point maps are studied from the viewpoint of dynamical systems. The structure of individual minimal sets and of the union M of all minimal sets of φ are investigated. One key result is that, if X is compact, then φ is a strongly almost periodic (i.e., has uniformly equicontinuous iterates) homeomorphisms of M and M is a retract of X. These and other general results are applied to the case where X is a manifold. Several results in which weak implies strong almost periodicity are obtained.     

Achievable sets, brambles, and sparse treewidth obstructions

One consequence of the graph minor theorem is that for every k there exists a finite obstruction set Obs(TW?k). However, relatively little is known about these sets, and very few general obstructions are known. The ones that are known are the cliques, and graphs which are formed by removing a few edges from a clique. This paper gives several general constructions of minimal forbidden minors which are sparse in the sense that the ratio of the treewidth to the number of vertices n does not approach 1 as n approaches infinity. We accomplish this by a novel combination of using brambles to provide lower bounds and achievable sets to demonstrate upper bounds. Additionally, we determine the exact treewidth of other basic graph constructions which are not minimal forbidden minors.     

Stability and extensibility results for abstract skew-product semiflows

In this paper we present new stability and extensibility results for skew-product semiflows with a minimal base flow. In particular, we describe the structure of uniformly stable and uniformly asymptotically stable sets admitting backwards orbits and the structure of omega-limit sets. As an application, the occurrence of almost periodic and almost automorphic dynamics for monotone non-autonomous infinite delay functional differential equations is analyzed.     

Determining Sets, Resolving Sets, and the Exchange Property

A subset U of vertices of a graph G is called a determining set if every automorphism of G is uniquely determined by its action on the vertices of U. A subset W is called a resolving set if every vertex in G is uniquely determined by its distances to the vertices of W. Determining (resolving) sets are said to have the exchange property in G if whenever S and R are minimal determining (resolving) sets for G and ${r\in R}$ , then there exists ${s\in S}$ so that ${S-\{s\} \cup \{r\}}$ is a minimal determining (resolving) set. This work examines graph families in which these sets do, or do not, have the exchange property. This paper shows that neither determining sets nor resolving sets have the exchange property in all graphs, but that both have the exchange property in trees. It also gives an infinite graph family (n-wheels where n ≥ 8) in which determining sets have the exchange property but resolving sets do not. Further, this paper provides necessary and sufficient conditions for determining sets to have the exchange property in an outerplanar graph.     

Nielsen equalizer theory

We extend the Nielsen theory of coincidence sets to equalizer sets, the points where a given set of (more than 2) mappings agree. On manifolds, this theory is interesting only for maps between spaces of different dimension, and our results hold for sets of k maps on compact manifolds from dimension (k−1)n to dimension n. We define the Nielsen equalizer number, which is a lower bound for the minimal number of equalizer points when the maps are changed by homotopies, and is in fact equal to this minimal number when the domain manifold is not a surface.As an application we give some results in Nielsen coincidence theory with positive codimension. This includes a complete computation of the geometric Nielsen number for maps between tori.