Sense Of Baire Category Research Articles

Topological prevalence of variable speed of convergence in the deterministic chaos game

Let A be the attractor of a Banach contractive iterated function system (IFS) on a complete space. We prove that the orbit generated by a typical (in the sense of Baire category) driver recovers A with every possible speed. Our result extends the one from the paper: Leśniak et al. (Chaos 32(1):013110, 2022). We also show that our result is optimal from a certain point of view.

Geodesic planes in a geometrically finite end and the halo of a measured lamination

Recent works [22,23,3,33] have shed light on the topological behavior of geodesic planes in the convex core of a geometrically finite hyperbolic 3-manifold M of infinite volume. In this paper, we focus on the remaining case of geodesic planes outside the convex core of M, giving a complete classification of their closures in M.In particular, we show that the behavior is different depending on whether exotic roofs exist or not. Here an exotic roof is a geodesic plane contained in an end E of M, which limits on the convex core boundary ∂E, but cannot be separated from the core by a support plane of ∂E.A necessary condition for the existence of exotic roofs is the existence of exotic rays for the bending lamination. Here an exotic ray is a geodesic ray that has a finite intersection number with a measured lamination L but is not asymptotic to any leaf nor eventually disjoint from L. We establish that exotic rays exist if and only if L is not a multicurve. The proof is constructive, and the ideas involved are important in the construction of exotic roofs.We also show that the existence of geodesic rays satisfying a stronger condition than being exotic, phrased only in terms of the hyperbolic surface ∂E and the bending lamination, is sufficient for the existence of exotic roofs. As a result, we show that geometrically finite ends with exotic roofs exist in every genus. Moreover, in genus 1, when the end is homotopic to a punctured torus, a generic one (in the sense of Baire category) contains uncountably many exotic roofs.

Typical Sequence of Real Numbers From the Unit Interval Has All Distribution Functions

This note is devoted to the study of typical properties (in Baire category sense) of sequences of real numbers in [0, 1]. We prove that the subset of sequences that have all distribution functions forms a residual set.

Generic uniformly continuous mappings on unbounded hyperbolic spaces

We consider a complete, unbounded hyperbolic metric space X and a concave, nonzero and nondecreasing function ω:[0,+∞)→[0,+∞) with ω(0)=0 and study the space Cω(X) of uniformly continuous self-mappings on X whose modulus of continuity is bounded from above by ω. We endow Cω(X) with the topology of uniform convergence on bounded sets and prove that the modulus of continuity of a generic mapping in Cω(X), in the sense of Baire categories, is precisely ω. Some related results in spaces of bounded mappings and in the topology of pointwise convergence are also discussed. This note can be seen as a completion of various results due to F. Strobin, S. Reich, A.J. Zaslavski, C. Bargetz and D. Thimm.

On universal quotient Blaschke products

In this paper, we introduce a new member to the universal families, called universal quotient Blaschke product, which is a formal quotient of two formal infinite Blaschke products. A formal infinite Blaschke product is of the form B ( z ) = ∏ k = 1 ∞ z − z k 1 − z k ¯ z , where { z k } k = 1 ∞ is a sequence of points in the unit disk but may not satisfy the Blaschke condition: ∑ k = 1 ∞ ( 1 − | z k | ) < ∞ . A partial quotient of a universal quotient Blaschke product is the quotient of two finite Blaschke products. We show that the set of partial quotients of a universal quotient Blaschke product is dense in the set of continuous self-mappings on the unit circle in the complex plane. Meanwhile, subsequences of the partial quotients of a universal quotient Blaschke product can be used to approximate any holomorphic functions bounded by one on the unit disk. Moreover, we prove that the set of universal quotient Blaschke products is huge in the sense of Baire category.

Generic uniqueness for the Plateau problem

Given a complete Riemannian manifold M⊂Rd which is a Lipschitz neighborhood retract of dimension m+n, of class Ch,β and an oriented, closed submanifold Γ⊂M of dimension m−1, which is a boundary in integral homology, we construct a complete metric space B of Ch,α-perturbations of Γ inside M, with α<β, enjoying the following property. For the typical element b∈B, in the sense of Baire categories, there exists a unique m-dimensional integral current in M which solves the corresponding Plateau problem and it has multiplicity one.

Generic uniqueness of optimal transportation networks

We prove that for the generic boundary, in the sense of Baire categories, there exists a unique minimizer of the associated optimal branched transportation problem.

Compact sets with large projections and nowhere dense sumset

We answer a question of Banakh, Jabłońska and Jabłoński by showing that for there exists a compact set such that the projection of K onto each hyperplane is of non-empty interior, but K + K is nowhere dense. The proof relies on a random construction. A natural approach in the proofs is to construct such a K in the unit cube with full projections, that is, such that the projections of K agree with that of the unit cube. We investigate the generalization of these problems for projections onto various dimensional subspaces as well as for -fold sumsets. We obtain numerous positive and negative results, but also leave open many interesting cases. We also show that in most cases if we have a specific example of such a compact set then actually the generic (in the sense of Baire category) compact set in a suitably chosen space is also an example. Finally, utilizing a computer-aided construction, we show that the compact set in the plane with full projections and nowhere dense sumset can be self-similar.

Generic length functions on countable groups

Let L(G) denote the space of integer-valued length functions on a countable group G endowed with the topology of pointwise convergence. Assuming that G does not satisfy any non-trivial mixed identity, we prove that a generic (in the Baire category sense) length function on G is a word length and the associated Cayley graph is isomorphic to a certain universal graph U independent of G. On the other hand, we show that every comeager subset of L(G) contains 2ℵ0 “asymptotically incomparable” length functions. A combination of these results yields 2ℵ0 pairwise non-equivalent regular representations G→Aut(U). We also prove that generic length functions are virtually indistinguishable from the model-theoretic point of view. Topological transitivity of the action of G on L(G) by conjugation plays a crucial role in the proof of the latter result.

A note on the interior and dimensions of typical sumsets

Let K \mathcal {K} be the collection of all non-empty compact subsets of the unit cube [ 0 , 1 ] d [0,1]^d . Fixing any F σ F_\sigma set F ⊂ [ 0 , 1 ] d F\subset [0,1]^d of Lebesgue measure zero and 0 &gt; c &gt; 1 0&gt;c&gt;1 , we show that for typical K ∈ K K\in \mathcal {K} with L d ( K ) ≥ c \mathcal {L}^d(K)\geq c (in the sense of Baire category), the sumset K + F K+F has empty interior. We also find the Hausdorff, lower and upper box and packing dimensions of K + F K+F for typical K ∈ K K\in \mathcal {K} .

On Successive Approximations for Compact-Valued Nonexpansive Mappings

We show that for a given initial point the typical, in the sense of Baire category, nonexpansive compact valued mapping F has the following properties: there is a unique sequence of successive approximations and this sequence converges to a fixed point of F. In the case of separable Banach spaces we show that for the typical mapping there is a residual set of initial points that have a unique trajectory.

Continuous functions with impermeable graphs

AbstractWe construct a Hölder continuous function on the unit interval which coincides in uncountably (in fact continuum) many points with every function of total variation smaller than 1 passing through the origin. We conclude that this function has impermeable graph—one of the key concepts introduced in this paper—and we present further examples of functions both with permeable and impermeable graphs. Moreover, we show that typical (in the sense of Baire category) continuous functions have permeable graphs. The first example function is subsequently used to construct an example of a continuous function on the plane which is intrinsically Lipschitz continuous on the complement of the graph of a Hölder continuous function with impermeable graph, but which is not Lipschitz continuous on the plane. As another main result, we construct a continuous function on the unit interval which coincides in a set of Hausdorff dimension 1 with every function of total variation smaller than 1 which passes through the origin.

Generic Properties of First-Order Mean Field Games

We consider a class of deterministic mean field games, where the state associated with each player evolves according to an ODE which is linear w.r.t. the control. Existence, uniqueness, and stability of solutions are studied from the point of view of generic theory. Within a suitable topological space of dynamics and cost functionals, we prove that, for “nearly all” mean field games (in the Baire category sense) the best reply map is single-valued for a.e. player. As a consequence, the mean field game admits a strong (not randomized) solution. Examples are given of open sets of games admitting a single solution, and other open sets admitting multiple solutions. Further examples show the existence of an open set of MFG having a unique solution which is asymptotically stable w.r.t. the best reply map, and another open set of MFG having a unique solution which is unstable. We conclude with an example of a MFG with terminal constraints which does not have any solution, not even in the mild sense with randomized strategies.

Existence and stability of weakly Pareto-Nash equilibria for discontinuous multiobjective games

We introduce two new concepts of the vector-valued generalized single deviation property and the vector-valued positive generalized single deviation property for discontinuous multiobjective games. Then, some new existence theorems of weakly Pareto-Nash equilibria for discontinuous multiobjective games are established. Moreover, we prove that most of the discontinuous multiobjective games are essential with perturbed payoffs in the sense of Baire category.

Recurrence properties for linear dynamical systems: An approach via invariant measures

We study different pointwise recurrence notions for linear dynamical systems from the Ergodic Theory point of view. We show that from any reiteratively recurrent vector x0, for an adjoint operator T on a separable dual Banach space X, one can construct a T-invariant probability measure which contains x0 in its support. This allows us to establish some equivalences, for these operators, between some strong pointwise recurrence notions which in general are completely distinguished. In particular, we show that (in our framework) reiterative recurrence coincides with frequent recurrence; for complex Hilbert spaces uniform recurrence coincides with the property of having a spanning family of unimodular eigenvectors; and the same happens for power-bounded operators on complex reflexive Banach spaces. These (surprising) properties are easily generalized to product and inverse dynamical systems, which implies some relations with the respective hypercyclicity notions. Finally we study how typical is an operator with a non-zero reiteratively recurrent vector in the sense of Baire category.

Baire category results for stochastic orders

In the sense of Baire categories, we prove that the elements of a typical pair of univariate distribution functions (defined on a bounded subset of {mathbb {R}}) cannot be compared in the sense of the usual stochastic order, the increasing convex order and the mean residual lifetime order. A similar result is also proved in the class of copulas, i.e. multivariate distribution functions with standard uniform marginals, equipped with the orthant order.

Optimal derivative loss for abstract wave equations

We consider an abstract wave equation with a propagation speed that depends only on time. We assume that the propagation speed is differentiable for positive times, continuous up to the origin, but with first derivative that is potentially singular at the origin. We examine the derivative loss of solutions, and in particular we investigate which conditions on the modulus of continuity and on the behavior of the derivative in the origin yield, respectively, no derivative loss, an arbitrarily small derivative loss, a finite derivative loss, or an infinite derivative loss. As expected, we obtain that stronger assumptions on the modulus of continuity can compensate weaker assumptions on the growth of the derivative, and viceversa. Suitable counterexamples show that our results are sharp. We prove indeed that, for every set of conditions, the class of propagation speeds that satisfy the given conditions, and for which the corresponding equation exhibits a derivative loss as large as possible, is nonempty and actually also residual in the sense of Baire category.

Generic power series on subsets of the unit disk

We examine the boundary behaviour of the generic power series f with coefficients chosen from a fixed bounded set Λ in the sense of Baire category. Notably, we prove that for any open subset U of the unit disk D with a nonreal boundary point on the unit circle, f(U) is a dense set of ℂ. As it is demonstrated, this conclusion does not necessarily hold for arbitrary open sets accumulating to the unit circle. To complement these results, a characterization of coefficient sets having this property is given.

Typicality results for weak solutions of the incompressible Navier–Stokes equations

In this work we show that, in the class of L∞((0,T); L2(T3)) distributional solutions of the incompressible Navier-Stokes system, the ones which are smooth in some open interval of times are meagre in the sense of Baire category, and the Leray ones are a nowhere dense set.

Does a typical ℓ_{𝑝}-space contraction have a non-trivial invariant subspace?

Given a Polish topology τ \tau on B 1 ( X ) \mathcal {B}_{1}(X) , the set of all contraction operators on X = ℓ p X=\ell _p , 1 ≤ p &gt; ∞ 1\le p&gt;\infty or X = c 0 X=c_0 , we prove several results related to the following question: does a typical T ∈ B 1 ( X ) T\in \mathcal {B}_{1}(X) in the Baire Category sense has a non-trivial invariant subspace? In other words, is there a dense G δ G_\delta set G ⊆ ( B 1 ( X ) , τ ) \mathcal G\subseteq (\mathcal {B}_{1}(X),\tau ) such that every T ∈ G T\in \mathcal G has a non-trivial invariant subspace? We mostly focus on the Strong Operator Topology and the Strong ∗ ^* Operator Topology.