Nonempty Interior Research Articles

Tropical fans and normal complexes: Putting the “volume” back in “volume polynomials”

Associated to any divisor in the Chow ring of a simplicial tropical fan, we construct a family of polytopal complexes, called normal complexes, which we propose as an analogue of the well-studied notion of normal polytopes from the setting of complete fans. We describe certain closed convex polyhedral cones of divisors for which the “volume” of each divisor in the cone—that is, the degree of its top power—is equal to the volume of the associated normal complexes. For the Bergman fan of any matroid with building set, we prove that there exists an open family of such cones of divisors with nonempty interiors. We view the theory of normal complexes developed in this paper as a polytopal model underlying the combinatorial Hodge theory pioneered by Adiprasito, Huh, and Katz.

Finite point configurations in products of thick Cantor sets and a robust nonlinear Newhouse Gap Lemma

AbstractIn this paper we prove that the set $\{|x^1-x^2|,\dots,|x^k-x^{k+1}|\,{:}\,x^i\in E\}$ has non-empty interior in $\mathbb{R}^k$ when $E\subset \mathbb{R}^2$ is a Cartesian product of thick Cantor sets $K_1,K_2\subset\mathbb{R}$ . We also prove more general results where the distance map $|x-y|$ is replaced by a function $\phi(x,y)$ satisfying mild assumptions on its partial derivatives. In the process, we establish a nonlinear version of the classic Newhouse Gap Lemma, and show that if $K_1,K_2, \phi$ are as above then there exists an open set S so that $\bigcap_{x \in S} \phi(x,K_1\times K_2)$ has non-empty interior.

The Baire theorem, an analogue of the Banach fixed point theorem and attractors in T1 compact spaces

We prove that if X is a T1 second countable compact space, then X is a Baire space if and only if every nonempty open subset of X contains a closed subset with nonempty interior. We also prove an analogue of Banach's fixed point theorem for all T1 compact spaces. Applying the analogue of Banach's fixed point theorem we prove the existence of unique attractors for so called contractive iterated function systems whose Hutchinson operators are closed in compact T1 spaces.

Inner and outer smooth approximation of convex hypersurfaces. When is it possible?

Let S be a convex hypersurface (the boundary of a closed convex set V with nonempty interior) in Rn. We prove that S contains no lines if and only if for every open set U⊃S there exists a real-analytic convex hypersurface SU⊂U∩int(V). We also show that S contains no rays if and only if for every open set U⊃S there exists a real-analytic convex hypersurface SU⊂U∖V. Moreover, in both cases, SU can be taken strongly convex. We also establish similar results for convex functions defined on open convex subsets of Rn, completely characterizing the class of convex functions that can be approximated in the C0-fine topology by smooth convex functions from above or from below. We also provide similar results for C1-fine approximations.

ON ARITHMETIC SUMS OF CONNECTED SETS IN

AbstractWe prove that for two connected sets $E,F\subset \mathbb {R}^2$ with cardinalities greater than $1$ , if one of E and F is compact and not a line segment, then the arithmetic sum $E+F$ has nonempty interior. This improves a recent result of Banakh et al. [‘The continuity of additive and convex functions which are upper bounded on non-flat continua in $\mathbb {R}^n$ ’, Topol. Methods Nonlinear Anal.54(1)(2019), 247–256] in dimension two by relaxing their assumption that E and F are both compact.

Convergence of iterates in nonlinear Perron-Frobenius theory

Let $ C $ be a closed cone with nonempty interior $ {{C}}^\circ $ in a Banach space. Let $ f:{{C}}^\circ \rightarrow {{C}}^\circ $ be an order-preserving subhomogeneous function with a fixed point in $ {{C}}^\circ $. We introduce a condition which guarantees that the iterates $ f^k(x) $ converge to a fixed point for all $ x \in {{C}}^\circ $. This condition generalizes the notion of type K order-preserving for maps on $ \mathbb{R}^n_{>0} $. We also prove that when iterates converge to a fixed point, the rate of convergence is always R-linear in two special cases: for piecewise affine maps and also for order-preserving, homogeneous, analytic, multiplicatively convex functions on $ \mathbb{R}^n_{>0} $. This later category includes the maps associated with the homogeneous eigenvalue problem for nonnegative tensors.

Typical self-affine sets with non-empty interior

Typical self-affine sets with non-empty interior

On the Lagrange and Markov dynamical spectra for geodesic flows on surfaces with negative curvature

We consider the Lagrange and the Markov dynamical spectra associated to a geodesic flow on a surface of negative curvature. We show that for a large set of real functions on the unit tangent bundle and for typical metrics with negative curvature and finite volume, both the Lagrange and the Markov dynamical spectra have non-empty interior.

Control Sets of Linear Control Systems on ℝ2. The Complex Case

This paper explicitly computes the unique control set D with the non-empty interior of a linear control system on ℝ2, when the associated matrix has complex eigenvalues. It turns out that the closure of D coincides with the region delimited by a computable periodic orbit 𝒪 of the system.

A Mattila-Sjölin theorem for triangles

We show for a compact set E⊂Rd, d≥4, that if the Hausdorff dimension of E is larger than 23d+1, then the set of congruence classes of triangles formed by triples of points of E has nonempty interior. Here we understand the set of congruence classes of triangles formed by triples of points of E as the setΔtri(E)={(t,r,α):|x−z|=t,|y−z|=r and Δtri(E)={,α=α(x,z,y),x,y,z∈E}, where α(x,z,y) denotes the angle formed by x, y and z, centered at z. This extends the Mattila-Sjölin theorem that establishes a non-empty interior for the distance set instead of the set of congruence classes of triangles. These theorems can be thought of as refinements and extensions of the statements in the well known Falconer distance problem.

On the convex components of a set in ℝ n

Abstract We prove a lower bound on the number of the convex components of a compact set with non-empty interior in ℝ n {\mathbb{R}^{n}} for all n ≥ 2 {n\geq 2} . Our result generalizes and improves the inequalities previously obtained in [M. Carozza, F. Giannetti, F. Leonetti and A. Passarelli di Napoli, Convex components, Commun. Contemp. Math. 21 2019, 6, Article ID 1850036] and [M. La Civita and F. Leonetti, Convex components of a set and the measure of its boundary, Atti Semin. Mat. Fis. Univ. Modena Reggio Emilia 56 2008/09, 71–78].

On stable and unstable behaviour of certain rotation segments

In this paper, we study non-wandering homeomorphisms of the two-dimen-sional torus homotopic to the identity, whose rotation sets are non-trivial segments from (0, 0) to some totally irrational point (α, β). We show that for any r ⩾ 1, this rotation set only appears for C r diffeomorphisms satisfying some degenerate conditions. And when such a rotation set does appear, assuming several natural conditions that are generically satisfied in the area-preserving world, we give a clearer description of its rotational behaviour. More precisely, the dynamics admits bounded deviation along the direction −(α, β) in the lift, and the rotation set is locked inside an arbitrarily small cone with respect to small C 0-perturbations of the dynamics. On the other hand, for any non-wandering homeomorphism f with this kind of rotation set, we also present a perturbation scheme in order for the rotation set to be eaten by the rotation set of some nearby dynamics, in the sense that the later set has non-empty interior and contains the former one. These two flavours interplay and share the common goal of understanding the stability/instability properties of this kind of rotation set.

Hypercyclic composition operators with automorphisms of the upper half-plane

On the Fréchet space H(P) of all holomorphic functions on the open upper half-plane P, we study universal sequences of composition operators Cσn:H(P)→H(P) given by automorphisms σn(z)=(anz+bn)/(cnz+dn) of P, where an,bn,cn,dn∈R with the normalization andn−bncn=1. We show that a sequence Cσn:H(P)→H(P) is universal if and only if limsup{|an|+|bn|+|cn|+|dn|}=∞, which in turn is equivalent to the existence of a point ζ in R∪{∞} such that some subsequence σnk(z)→ζ uniformly on compact subsets of P. Applying these conditions to the case when each σn is the n-fold composition σn(z)=σ∘⋯∘σ(z) of a non-identity automorphism σ(z)=(az+b)/(cz+d), where a,b,c,d∈R with the normalization ad−bc=1, we show that Cσ is hypercyclic if and only if |a+d|≥2. Furthermore we obtain an explicit formula, in terms of a,b,c,d, of the point ζ for which σn(z)→ζ uniformly on compact subsets of P. Motivated by our results for the region P, we obtain analogous results when the region is the open unit disk. Finally we generalize the aforementioned limit-point result to hypercyclic composition operators with automorphisms on a simply connected region whose complement has a nonempty interior.

On families of constrictions in model of overdamped Josephson junction and Painlevé 3 equation* *Supported by RSF Grant 18-41-05003.

The tunnelling effect predicted by Josephson (Nobel Prize, 1973) concerns the Josephson junction: two superconductors separated by a narrow dielectric. It states existence of a supercurrent through it and equations governing it. The overdamped Josephson junction is modelled by a family of differential equations on two-torus depending on three parameters: B (abscissa), A (ordinate), ω (frequency). We study its rotation number ρ(B, A; ω) as a function of (B, A) with fixed ω. The phase-lock areas are the level sets L r ≔ {ρ = r} with non-empty interiors; they exist for (Buchstaber, Karpov, Tertychnyi). Each L r is an infinite chain of domains going vertically to infinity and separated by points. Those separating points for which A ≠ 0 are called constrictions. We show that: (1) all the constrictions in L r lie on the axis {B = ωr}; (2) each constriction is positive: this means that some its punctured neighbourhood on the axis {B = ωr} lies in Int(L r ). These results confirm experiments by physicists (1970ths) and two mathematical conjectures. We first prove deformability of each constriction to another one, with arbitrarily small ω and the same , using equivalent description of model by linear systems of differential equations on (Buchstaber, Karpov, Tertychnyi) and studying their isomonodromic deformations described by Painlevé 3 equations. Then non-existence of ghost constrictions (i.e., constrictions either with , or of non-positive type) with a given ℓ for small ω is proved by slow-fast methods.

On the Mattila–Sjölin distance theorem for product sets

Let A be a compact set in R $\mathbb {R}$ , and E = A d ⊂ R d $E=A^d\subset \mathbb {R}^d$ . We know from the Mattila–Sjölin's theorem if dim H ( A ) > d + 1 2 d $\dim _H(A)>\frac{d+1}{2d}$ , then the distance set Δ ( E ) $\Delta (E)$ has non-empty interior. In this paper, we show that the threshold d + 1 2 d $\frac{d+1}{2d}$ can be improved whenever d ⩾ 5 $d\geqslant 5$ .

The Walrasian objection mechanism and Mas-Colell’s bargaining set in economies with many commodities

We study the Walrasian objection mechanism in the framework of economies with a measure space of agents and a separable Banach space of commodities whose positive cone has a non-empty interior. We provide several characterizations of Walrasian objections and use them to study the bargaining set of the economy, as defined in Mas-Colell (J Math Econ 18(2):129–139, 1989). Our main result shows that whenever the measure space of agents is saturated, every non-competitive allocation can be blocked with a Walrasian objection. This implies that the bargaining set, the core and the set of competitive allocations are equivalent solution concepts.

Approximate solutions for set optimization with an order cone that has nonempty quasirelative interiors

In a real normed linear space, when the quasirelative interior is not empty, a class of order relation is introduced with Minkowski difference. Two classes of nonlinear functions are introduced, and their properties are discussed. A class of approximately efficient solutions and approximate weakly efficient solutions are introduced for set optimization. With nonlinear functions, optimality conditions are established for approximate solutions. Some examples are given to illustrate our main results.

On the topological structure of the (non-)finitely generated locus of Frobenius algebras emerging from Stanley–Reisner rings

Let U be the set of prime ideals P of the completion of a Stanley–Reisner ring S, such that the localization at P of the Frobenius algebra of the injective hull of the residue field of S is a finitely generated algebra. We give a partial answer to a conjecture made by M. Katzman about the openness of U. Specifically, we show that U has non-empty interior and we present some sufficient conditions for a principal open set D(f) to be contained in U, and for intersections of closed and principal opens to be contained in the complement of U.

Measurable equidecompositions for group actions with an expansion property

Given an action of a group \Gamma on a measure space \Omega , we provide a sufficient criterion under which two sets A, B\subset \Omega are \textit{measurably equidecomposable} , i.e., A can be partitioned into finitely many measurable pieces which can be rearranged using some elements of \Gamma to form a partition of B . In particular, we prove that every bounded measurable subset of \mathbb{R}^n , n\ge 3 , with non-empty interior is measurably equidecomposable to a ball via isometries. The analogous result also holds for some other spaces, such as the sphere or the hyperbolic space of dimension n\ge 2 .

The absolute continuity of convolutions of orbital measures in $$SO(2n+1)$$

Let G be a compact Lie group of Lie type $$B_{n},$$ such as $$SO(2n+1)$$ . We characterize the tuples $$(x_{1},\ldots ,x_{L})$$ of the elements $$x_{j}\in G$$ which have the property that the product of their conjugacy classes has non-empty interior. Equivalently, the convolution product of the orbital measures supported on their conjugacy classes is absolutely continuous with respect to Haar measure. The characterization depends on the dimensions of the largest eigenspaces of each $$x_{j}$$ . Such a characterization was previously only known for the compact Lie groups of type $$A_{n}$$ .