On minimal non-potentially closed subsets of the plane

We study the Borel subsets of the plane that can be made closed by refining the Polish topology on the real line. These sets are called potentially closed. We first compare Borel subsets of the plane using products of continuous functions. We show the existence of a perfect antichain made of minimal sets among non-potentially closed sets. We apply this result to graphs, quasi-orders and partial orders. We also give a non-potentially closed set minimum for another notion of comparison. Finally, we show that we cannot have injectivity in the Kechris-Solecki-Todor?evi? dichotomy about analytic graphs.     

Measure of minimal sets for certain discrete dynamical systems

Let X be a separable metric space, μ a complete Borel measure on X that is finite on balls, and f a closed discrete dynamical system on X that preserves μ and has the diameters of all orbits bounded. We prove that almost every point in X (in the sense of measure μ) has its orbit contained in its ω-limit set.     

Ergodic properties for some non-expanding non-reversible systems

We study several properties of invariant measures obtained from preimages, for non-invertible maps on fractal sets which model non-reversible dynamical systems. We give two ways to describe the distribution of all preimages for endomorphisms which are not necessarily expanding on a basic set Λ. We give a topological dynamics condition which guarantees that the corresponding measures converge to a unique conformal ergodic borelian measure; this helps in estimating the unstable dimension a.e. with respect to this measure with the help of Lyapunov exponents. When there exist negative Lyapunov exponents of this limit measure, we study the conditional probabilities induced on the non-uniform local stable manifolds by the limit measure, and also its pointwise dimension on stable manifolds.     

Forests and score vectors

Thescore vector of a labeled digraph is the vector of out-degrees of its vertices. LetG be a finite labeled undirected graph without loops, and let σ(G) be the set of distinct score vectors arising from all possible orientations ofG. Let ϕ(G) be the set of subgraphs ofG which are forests of labeled trees. We display a bijection between σ(G) and ϕ(G). Supported in part by ONR Contract N00014-76-C-0366.     

On Cocycles with Values in the Group SU(2)

 In this paper we introduce the notion of degree for C ¹-cocycles over irrational rotations on the circle with values in the group SU(2). It is shown that if a C ¹-cocycle over an irrational rotation by α has nonzero degree, then the skew product

Cantor families of periodic solutions for wave equations via a variational principle

We prove existence of small amplitude periodic solutions of completely resonant wave equations with frequencies in a Cantor set of asymptotically full measure, via a variational principle. A Lyapunov-Schmidt decomposition reduces the problem to a finite dimensional bifurcation equation—variational in nature—defined on a Cantor set of non-resonant parameters. The Cantor gaps are due to “small divisors” phenomena. To solve the bifurcation equation we develop a suitable variational method. In particular, we do not require the typical “Arnold non-degeneracy condition” of the known theory on the nonlinear terms. As a consequence our existence results hold for new generic sets of nonlinearities.     

On negative limit sets for one-dimensional dynamics

In this paper we study the structure of negative limit sets of maps on the unit interval. We prove that every α-limit set is an ω-limit set, while the converse is not true in general. Surprisingly, it may happen that the space of all α-limit sets of interval maps is not closed in the Hausdorff metric (and thus some ω-limit sets are never obtained as α-limit sets). Moreover, we prove that the set of all recurrent points is closed if and only if the space of all α-limit sets is closed.     

Almost disjoint large subsets of semigroups

There are several notions of largeness in a semigroup S that originated in topological dynamics. Among these are thick, central, syndetic and piecewise syndetic. Of these, central sets are especially interesting because they are partition regular and are guaranteed to contain substantial combinatorial structure. It is known that in (N,+) any central set may be partitioned into infinitely many pairwise disjoint central sets. We extend this result to a large class of semigroups (including (N,+)) by showing that if S is a semigroup in this class which has cardinality κ then any central set can be partitioned into κ many pairwise disjoint central sets. We also show that for this same class of semigroups, if there exists a collection of μ almost disjoint subsets of any member S, then any central subset of S contains a collection of μ almost disjoint central sets. The same statement applies if “central” is replaced by “thick”; and in the case that the semigroup is left cancellative, “central” may be replaced by “piecewise syndetic”. The situation with respect to syndetic sets is much more restrictive. For example, there does not exist an uncountable collection of almost disjoint syndetic subsets of N. We investigate the extent to which syndetic sets can be split into disjoint syndetic sets.     

On nonmeasurable unions

For a Polish space X and a σ-ideal I of subsets of X which has a Borel base we consider families A of sets in I with the union ?A not in I. We determine several conditions on A which imply the existence of a subfamily A^′ of A whose union ?A^′ is not in the σ-field generated by the Borel sets on X and I. Main examples are X=R and I being the ideal of sets of Lebesgue measure zero or the ideal of sets of the first Baire category.     

A Fubini theorem

Let I₀ be the σ-ideal of subsets of a Polish group generated by Borel sets which have perfectly many pairwise disjoint translates. We prove that a Fubini-type theorem holds between I₀ and the σ-ideals of Haar measure zero sets and of meager sets. We use this result to give a simple proof of a generalization of a theorem of Balcerzak-Ros?anowski-Shelah stating that I₀ on N² strongly violates the countable chain condition.     

Harmonic Functions on Homogeneous Spaces

 Given a locally compact group G acting on a locally compact space X and a probability measure σ on G, a real Borel function f on X is called σ-harmonic if it satisfies the convolution equation . We give conditions for the absence of nonconstant bounded harmonic functions. We show that, if G is a union of σ-admissible neighbourhoods of the identity, relative to X, then every bounded σ-harmonic function on X is constant. Consequently, for spread out σ, the bounded σ-harmonic functions are constant on each connected component of a [SIN]-group and, if G acts strictly transitively on a splittable metric space X, then the bounded σ-harmonic functions on X are constant which extends Furstenberg’s result for connected semisimple Lie groups. (Received 13 June 1998; in revised form 31 March 1999)     

Matrix orthogonal Laurent polynomials on the unit circle and Toda type integrable systems

Matrix orthogonal Laurent polynomials in the unit circle and the theory of Toda-like integrable systems are connected using the Gauss–Borel factorization of two, left and a right, Cantero–Morales–Velázquez block moment matrices, which are constructed using a quasi-definite matrix measure. A block Gauss–Borel factorization problem of these moment matrices leads to two sets of biorthogonal matrix orthogonal Laurent polynomials and matrix Szeg? polynomials, which can be expressed in terms of Schur complements of bordered truncations of the block moment matrix. The corresponding block extension of the Christoffel–Darboux theory is derived. Deformations of the quasi-definite matrix measure leading to integrable systems of Toda type are studied. The integrable theory is given in this matrix scenario; wave and adjoint wave functions, Lax and Zakharov–Shabat equations, bilinear equations and discrete flows — connected with Darboux transformations. We generalize the integrable flows of the Cafasso's matrix extension of the Toeplitz lattice for the Verblunsky coefficients of Szeg? polynomials. An analysis of the Miwa shifts allows for the finding of interesting connections between Christoffel–Darboux kernels and Miwa shifts of the matrix orthogonal Laurent polynomials.     

Measurable recurrence and quasi-invariant measures

We characterize the wandering sets of a Borel automorphismT as being precisely those sets which have measure zero for every non-atomic measure μ which is quasi-invariant and ergodic forT.     

On graphs whose star sets are (co-)cliques

In this paper we study graphs all of whose star sets induce cliques or co-cliques. We show that the star sets of every tree for each eigenvalue are independent sets. Among other results it is shown that each star set of a connected graph G with three distinct eigenvalues induces a clique if and only if G=K_1,2 or K_2,…,2. It is also proved that stars are the only graphs with three distinct eigenvalues having a star partition with independent star sets.     

Invariant measures for equicontinuous semigroups of continuous transformations of a compact Hausdorff space

Equicontinuous semigroups of transformations of a compact Hausdorff space and their sets of all invariant (Borel, regular and probabilistic) measures are studied. Conditions equivalent to the existence of at least one invariant measure are given. The (algebraic and topological) structure of the set of invariant measures is researched.     

Multipartite Moore digraphs

We derive some Moore-like bounds for multipartite digraphs, which extend those of bipartite digraphs, under the assumption that every vertex of a given partite set is adjacent to the same number δ of vertices in each of the other independent sets. We determine when a multipartite Moore digraph is weakly distance-regular. Within this framework, some necessary conditions for the existence of a r-partite Moore digraph with interpartite outdegree δ > 1 and diameter k = 2m are obtained. In the case δ = 1, which corresponds to almost Moore digraphs, a necessary condition in terms of the permutation cycle structure is derived. Additionally, we present some constructions of dense multipartite digraphs of diameter two that are vertex-transitive.     

Lefschetz fixed point theory on fibrations

For a fibre preserving map ϕ: E → E on a fibration (E, π, B), we construct a grading preserving map T(ϕ, π) between H*(E) and H*(B) that generalizes the Lefschetz number. If T(ϕ, π) is an isomorphism between H ⁰(E) and H ⁰(B), then π restricts to a surjective local diffeomorphism on each connected component of the fixed point set of ϕ under a transversality condition. This yields a characterization for the bundle H → G → G/H to be trivial when π ₁ (G/H) = 0.     

An example on movable approximations of a minimal set in a continuous flow

In the present paper the study of flows on n-manifolds in particular in dimension three, e.g., R³, is motivated by the following question. Let A be a compact invariant set in a flow on X. Does every neighbourhood of A contain a movable invariant set M containing A? It is known that a stable solenoid in a flow on a 3-manifold has approximating periodic orbits in each of its neighbourhoods. The solenoid with the approximating orbits form a movable set, although the solenoid is not movable. Not many such examples are known. The main part of the paper consists of constructing an example of a set in R³ that is not stable, is not a solenoid, and is approximated by Denjoy-like invariant sets instead of periodic orbits. As in the case of a solenoid, the constructed set is an inverse limit of its approximating sets. This gives a partial answer to the above question.