Nonempty Open Set Research Articles

Some generic properties of Schrödinger operators with radial potentials

AbstractWe consider a class of Schrödinger operators on ${\open R}^N$ with radial potentials. Viewing them as self-adjoint operators on the space of radially symmetric functions in $L^2({\open R}^N)$, we show that the following properties are generic with respect to the potential: (P1)the eigenvalues below the essential spectrum are nonresonant (i.e., rationally independent) and so are the square roots of the moduli of these eigenvalues;(P2)the eigenfunctions corresponding to the eigenvalues below the essential spectrum are algebraically independent on any nonempty open set. The genericity means that in suitable topologies the potentials having the above properties form a residual set. As we explain, (P1), (P2) are prerequisites for some applications of KAM-type results to nonlinear elliptic equations. Similar properties also play a role in optimal control and other problems in linear and nonlinear partial differential equations.

Coloring Cantor sets and resolvability of pseudocompact spaces

Let us denote by $\Phi(\lambda,\mu)$ the statement that $\mathbb{B}(\lambda) = D(\lambda)^\omega$, i.e. the Baire space of weight $\lambda$, has a coloring with $\mu$ colors such that every homeomorphic copy of the Cantor set $\mathbb{C}$ in $\mathbb{B}(\lambda)$ picks up all the $\mu$ colors. We call a space $X$ $\pi$-regular if it is Hausdorff and for every nonempty open set $U$ in $X$ there is a nonempty open set $V$ such that $\overline{V} \subset U$. We recall that a space $X$ is called feebly compact if every locally finite collection of open sets in $X$ is finite. A Tychonov space is pseudocompact if and only if it is feebly compact. The main result of this paper is the following: Let $X$ be a crowded feebly compact $\pi$-regular space and $\mu$ be a fixed (finite or infinite) cardinal. If $\Phi(\lambda,\mu)$ holds for all $\lambda < \hat{c}(X)$ then $X$ is $\mu$-resolvable, i.e. $X$ contains $\mu$ pairwise disjoint dense subsets. (Here $\hat{c}(X)$ is the smallest cardinal $\kappa$ such that $X$ does not contain $\kappa$ many pairwise disjoint open sets.) This significantly improves earlier results of [van Mill J., {Every crowded pseudocompact ccc space is resolvable}, Topology Appl. 213 (2016), 127--134], or [Ortiz-Castillo Y. F., Tomita A. H., {Crowded pseudocompact Tychonoff spaces of cellularity at most the continuum are resolvable}, Conf. talk at Toposym 2016].

On the maximal rank problem for the complex homogeneous Monge–Ampère equation

We give examples of regular boundary data for the Dirichlet problem for the Complex Homogeneous Monge-Amp\`ere Equation over the unit disc, whose solution is completely degenerate on a non-empty open set and thus fails to have maximal rank.

Approximate Controllability from the Exterior of Space-Time Fractional Diffusive Equations

Let $\Omega\subset\mathbb{R}^N$ be a bounded domain with a Lipschitz continuous boundary. We study the controllability of the space-time fractional diffusive equation $\{\mathbb D_t^\alpha u+(-\Delta)^su=0 \mbox{ in}\;(0,T)\times\Omega, u=g\chi_{(0,T)\times\mathcal O} %& \mbox{ in}\;(0,T)\times(\mathbb{R}^N\setminus\Omega), u(0,\cdot)=u_0%& \mbox{ in}\;\Omega\}$, where $u=u(t,x)$ is the state to be controlled and $g=g(t,x)$ is the control function which is localized in a nonempty open subset $\mathcal O$ of $\mathbb{R}^N\setminus\Omega$. Here, $0<\alpha\le 1$, $0<s<1,$ and $T>0$ are real numbers. After giving an explicit representation of solutions, we show that the system is always approximately controllable for every $T>0$, $u_0\in L^2(\Omega)$, and $g\in \mathcal D((0,T)\times\mathcal O),$ where $\mathcal O\subseteq\mathbb{R}^N\setminus\Omega$ is an arbitrary nonempty open set. The results obtained are sharp in the sense that such a system is never null controllable if $0<\alpha<1$. The proof of our result is based on a new unique continuation principle for the eigenvalues problem associated with the fractional Laplace operator subject to the zero Dirichlet exterior condition that we have established.

A study of cellular-Lindelöf spaces

The class of cellular-Lindelöf spaces was introduced and studied by A. Bella and S. Spadaro (2017) [4]. We say that a topological space X is cellular-Lindelöf if for every family U of pairwise disjoint non-empty open sets of X there is a Lindelöf subspace L⊂X such that U∩L≠∅, for every U∈U. In this paper, we first study topological properties of cellular-Lindelöf spaces, and the relations between cellular-Lindelöf spaces and related spaces. In particular, we obtain a Tychonoff example of a weakly Lindelöf space which is not cellular-Lindelöf, which gives a positive answer to a question of A. Bella and S. Spadaro ([4, Question 2]). We also prove that every monotonically normal W-space is cellular-Lindelöf if and only if it is Lindelöf. Finally, by using Erdös–Radó's theorem, we establish some cardinal inequalities for cellular-Lindelöf spaces. Some new questions are also posed.

Double minimality, entropy and disjointness with all minimal systems

In this paper we propose a new sufficient condition for disjointness with all minimal systems. Using proposed approach we construct a transitive dynamical system \begin{document}$(X,T)$ \end{document} disjoint with every minimal system and such that the set of transfer times \begin{document}$N(x,U)$ \end{document} is not an \begin{document}$\text{IP}^*$ \end{document} -set for some nonempty open set \begin{document}$U\subset X$ \end{document} and every \begin{document}$x∈ X$ \end{document} . This example shows that the new condition sharpens sufficient conditions for disjointness below previous bounds. In particular our approach does not rely on distality of points or sets.

Fine topology and locally Minkowskian manifolds

Fine topology is one of the several well-known topologies of physical and mathematical relevance. In the present paper, it is obtained that the nonempty open sets of different dimensional Minkowski spaces with the fine topology are not homeomorphic. This leads to the introduction of a new class of manifolds. It turns out that the technique developed here is also applicable to some other topologies, namely, the s-topology, space topology, f-topology, and A-topology.

Frequently hypercyclic operators with irregularly visiting orbits

We prove that a bounded operator T on a separable Banach space X satisfying a strong form of the Frequent Hypercyclicity Criterion (which implies in particular that the operator is universal in the sense of Glasner and Weiss) admits frequently hypercyclic vectors with irregularly visiting orbits, i.e. vectors x∈X such that the set NT(x,U)={n≥1;Tnx∈U} of return times of x into U under the action of T has positive lower density for every non-empty open set U⊆X, but there exists a non-empty open set U0⊆X such that NT(x,U0) has no density.

An Almost Everywhere Constant Function Surjective on Every Interval

We describe an almost everywhere constant function surjective on every nonempty open set.

Topological properties of a curved spacetime

The present paper aims at the study of a topology on Lorentzian manifolds, defined by Gobel [ 4 ] using the ideas of Zeeman [ 16 ]. Observing that on the Minkowski space it is the same as Zeeman's time topology, it has been found that a Lorentzian manifold with this topology is path connected, nonfirst countable and nonsimply connected while the Minkowski space with time topology is, in addition nonregular and separable. Furthermore, using the notion of Zeno sequences it is obtained that a compact set does not contain a nonempty open set and that a set is compact if and only if each of its infinite subsets has a limit point if and only if each of its sequences has a convergent subsequence.

When is multiplication in a Banach algebra open?

We develop the theory of Banach algebras whose multiplication (regarded as a bilinear map) is open. We demonstrate that such algebras must have topological stable rank 1, however the latter condition is strictly weaker and implies only that products of non-empty open sets have non-empty interior. We then investigate openness of convolution in semigroup algebras resolving in the negative a problem of whether convolution in ℓ1(N0) is open. By appealing to ultraproduct techniques, we demonstrate that neither in ℓ1(Z) nor in ℓ1(Q) convolution is uniformly open. The problem of openness of multiplication in Banach algebras of bounded operators on Banach spaces and their Calkin algebras is also discussed.

Exceptional values of holomorphic functions. Remarks on a Nishino's Theorem

Let F(z,w)∈O(Cn+1), where (z,w)∈Cn×C. Let z′∈Cn such that F(z′,w) is not constant. If F(z′,w) is not surjective it takes all the values of C minus one π(z′) (Picard). T. Nishino studied in [8]π(z) when n=1, F(z,w) is of finite order in w and π(z) is defined in a set E⊂C with at least one accumulation point. In this work, we see that his result allows to obtain an explicit expression of such a F(z,w) when n≥1 and F(z′,w) is not a constant for any z′∈Cn, and conclude that π(z)=η(z)−1/ξ(z) for η(z) and ξ(z)∈O(Cn) when π(z) is defined on a nonempty open set U⊂Cn. Moreover, we give several applications of this fact. We show that the complement of the graph of π(z) in Cn+1 is dominated by Cn+1 via a family of surjective fiber-preserving holomorphic maps with non-vanishing Jacobian determinant, which are described in terms of the flow of a complete vector field of type C⁎. In particular, Buzzard and Lu's results in [2] applied to π(z) for n=1 can be extended for n≥2. It will allow to define new examples of Oka manifolds.

On topological symplectic dynamical systems

This paper contributes to the study of topological symplectic dynamical systems, and hence to the extension of smooth symplectic dynamical systems. Using the positivity result of symplectic displacement energy [4], we prove that any generator of a strong symplectic isotopy uniquely determine the latter. This yields a symplectic analogue of a result proved by Oh [12], and the converse of the main theorem found in [6]. Also, tools for defining and for studying the topological symplectic dynamical systems are provided: We construct a right-invariant metric on the group of strong symplectic homeomorphisms whose restriction to the group of all Hamiltonian homeomorphism is equivalent to Oh’s metric [12], define the topological analogues of the usual symplectic displacement energy for non-empty open sets, and we prove that the latter is positive. Several open conjectures are elaborated.

Local rearrangement invariant spaces and distribution of Rademacher series

We prove that a local version of Khintchine inequality holds for arbitrary rearrangement invariant (r.i.) spaces on an non-empty open set $$E\subset [0,1]$$ . For this, we give a definition of local r.i. space which is compatible with the notion of systems equivalent in distribution and prove that the Rademacher system $$(r_{k+N})_{k=1}^\infty $$ on an non-empty open set E is equivalent in distribution to $$(r_k)_{k=1}^\infty $$ on [0, 1], with N depending on E. The result can be generalized to a wider class of sets.

Transitive and series transitive maps on [formula omitted

Motivated by the behavior of topologically transitive homomorphisms of Polish abelian groups, we say a continuous map f:Rd→Rd is ‘series transitive’ if for any two nonempty open sets U,V⊂Rd, there exist x∈U and n∈N such that ∑j=0n−1fj(x)∈V. We show that any map on a discrete and closed subset of Rd can be extended to a mixing map of Rd, and use this result to produce a mixing map f:Rd→Rd (for each d∈N) which is also series transitive. We have examples to say that transitivity and series transitivity are independent properties for continuous self-maps of Rd. We also construct a chaotic map (i.e., a transitive map with a dense set of periodic points) f:Rd→Rd such that f is arbitrarily close to and asymptotic to the identity map. Finally, we make a few observations about topological transitivity of continuous homomorphisms of Polish abelian groups.

Control of eigenfunctions on hyperbolic surfaces: an application of fractal uncertainty principle

This expository article, written for the proceedings of the Journ\'ees EDP (Roscoff, June 2017), presents recent work joint with Jean Bourgain [arXiv:1612.09040] and Long Jin [arXiv:1705.05019]. We in particular show that eigenfunctions of the Laplacian on hyperbolic surfaces are bounded from below in $L^2$ norm on each nonempty open set, by a constant depending on the set but not on the eigenvalue.

ISOMETRIC REPRESENTATION OF LIPSCHITZ‐FREE SPACES OVER CONVEX DOMAINS IN FINITE‐DIMENSIONAL SPACES

Let be a finite-dimensional normed space and a non-empty convex open set in . We show that the Lipschitz-free space of is canonically isometric to the quotient of by the subspace consisting of vector fields with zero divergence in the sense of distributions on .

Rectifiability of a class of invariant measures with one non-vanishing Lyapunov exponent

We study order-preserving $\mathcal{C}^1$-circle diffeomorphisms driven by irrational rotations with a Diophantine rotation number. We show that there is a non-empty open set of one-parameter families of such diffeomorphisms where the ergodic measures of nearly all family members are one-rectifiable, that is, absolutely continuous with respect to the restriction of the one-dimensional Hausdorff measure to a countable union of Lipschitz graphs.

Volume hyperbolicity and wildness

It is known that volume hyperbolicity (partial hyperbolicity and uniform expansion or contraction of the volume in the extremal bundles) is a necessary condition for robust transitivity or robust chain recurrence and hence for tameness. In this paper, on any $3$-manifold we build examples of quasi-attractors which are volume hyperbolic and wild at the same time. As a main corollary, we see that, for any closed $3$-manifold $M$, the space $\text{Diff}^{1}(M)$ admits a non-empty open set where every $C^{1}$-generic diffeomorphism has no attractors or repellers. The main tool of our construction is the notion of flexible periodic points introduced in the authors’ previous paper. In order to eject the flexible points from the quasi-attractor, we control the topology of the quasi-attractor using the notion of partially hyperbolic filtrating Markov partitions, which we introduce in this paper.

Openly Haar null sets and conjugacy in Polish groups

This paper is motivated by the question of Darji: May every uncountable Polish group be written as the disjoint union G = A∪B where A is a Haar null set and B is meager? We give some weak criteria which answer the question in the affirmative for at least the following groups: any TSI group, any countably infinite product of locally compact groups, S∞, Aut(Q,≤), U(H), and Diff+k[0, 1]. Our methods involve a study of the connection between the existence of distal pairs with respect to a Polish group G’s action by conjugation on itself, and the size of the group’s openly Haar null ideal OHN(G) as identified by Solecki, and we give a smattering of results regarding the scope of this ideal in groups both locally compact and not. In the group G = Homeo+[0, 1] of orientation-preserving self-homeomorphisms of the interval, we exhibit the following curious two-sided translation property, impossible in any (non-trivial) locally compact group: for every compact set K ⊆ G and every nonempty open set U ⊆ K there are g, h ∈ G with gKh ⊆ U. Accordingly, the openly Haar null ideal OHN(Homeo+[0, 1]) is trivial. Lastly, in the appendix we identify and correct a small error in a frequently cited proof of Mycielski that a countable union of Haar null sets is Haar null in any Polish group G.