Sense Of Baire Research Articles

A Typical Number is Extremely Non-Normal

Abstract Fix a positive integer N ≥ 2. For a real number x ∈ [0, 1] and a digit i ∈ {0, 1,..., N − 1}, let Π i (x, n) denote the frequency of the digit i among the first nN-adic digits of x. It is well-known that for a typical (in the sense of Baire) x ∈ [0, 1], the sequence of digit frequencies diverges as n →∞. In this paper we show that for any regular linear transformation T there exists a residual set of points x ∈ [0,1] such that the T -averaged version of the sequence (Π i (x, n)) n also diverges significantly.

Generic uniqueness of saddle point for two-person zero-sum differential games

Abstract The generic uniqueness of saddle point for two-person zero-sum differential games, within the class of open-loop, against the perturbation of the right-hand side function of the control system is investigated. By employing set-valued mapping theory, it is proved that the majority of the two-person zero-sum differential games have unique saddle point in the sense of Baire’s category.

Existence of minimal hypersurfaces with non-empty free Boundary for generic metrics

For almost all Riemannian metrics (in the $C^\infty$ Baire sense) on a compact manifold with boundary $(M^{n+1},\partial M)$, $3\leq (n + 1)\leq 7$, we prove that, for any open subset $V$ of $\partial M$, there exists a compact, properly embedded free boundary minimal hypersurface intersecting $V$.

Stability and prevalence of McKean–Vlasov stochastic differential equations with non-Lipschitz coefficients

Abstract We consider various approximation properties for systems driven by a McKean–Vlasov stochastic differential equations (MVSDEs) with continuous coefficients, for which pathwise uniqueness holds. We prove that the solution of such equations is stable with respect to small perturbation of initial conditions, parameters and driving processes. Moreover, the unique strong solutions may be constructed by an effective approximation procedure. Finally, we show that the set of bounded uniformly continuous coefficients for which the corresponding MVSDE have a unique strong solution is a set of second category in the sense of Baire.

Min\u2013max theory for free boundary minimal hypersurfaces II: general Morse index bounds and applications

For any smooth Riemannian metric on an (n+1)-dimensional compact manifold with boundary (M,partial M) where 3le (n+1)le 7, we establish general upper bounds for the Morse index of free boundary minimal hypersurfaces produced by min–max theory in the Almgren–Pitts setting. We apply our Morse index estimates to prove that for almost every (in the C^infty Baire sense) Riemannan metric, the union of all compact, properly embedded free boundary minimal hypersurfaces is dense in M. If partial M is further assumed to have a strictly mean convex point, we show the existence of infinitely many compact, properly embedded free boundary minimal hypersurfaces whose boundaries are non-empty. Our results prove a conjecture of Yau for generic metrics in the free boundary setting.

Average frequencies of digits in infinite IFS\u2019s and applications to continued fractions and L\xfcroth expansions

The detailed investigation of the distribution of frequencies of digits of points belonging to attractors K of Infinite iterated functions systems (IIFS’s) is a fundamental and important problem in the study of attractors of IIFS’s. This paper studies the Baire category of different families of sets of points belonging to attractors of IIFS’s characterised by the behaviour of the frequencies of their digits. All our results are of the following form:a typical (in the sense of Baire) point xin K has the following property: the average frequencies of digits of x have maximal oscillation.We consider general types of average frequencies, namely, average frequencies associated with general averaging systems. These averages include, for example, all higher order Hölder and Cesaro averages, and Riesz averages. Surprising, for all averaging systems (regardless of how powerful they are) we prove that a typical (in the sense of Baire) point xin K has the following property: the average frequencies of digits of x have maximal oscillation. This substantially extends previous results and provides a powerful topological manifestation of the fact that “points of divergence” are highly visible. Several applications are given, e.g. to continued fraction digits and Lüroth expansion digits.

Generic stability of Nash equilibria for noncooperative differential games

The stability of Nash equilibria against the perturbation of the right-hand side functions of state equations for noncooperative differential games is investigated. By employing the set-valued analysis theory, we show that the differential games whose equilibria are all stable form a dense residual set, and every differential game can be approximated arbitrarily by a sequence of stable differential games, that is, in the sense of Baire’s category most of the differential games are stable.

Spectrum of the Landau Hamiltonian with a Periodic Electric Potential

We define a class of periodic electric potentials for which the spectrum of the two-dimensional Schrodinger operator is absolutely continuous in the case of a homogeneous magnetic field B with a rational flux η = (2π)−1Bυ(K), where υ(K) is the area of an elementary cell K in the lattice of potential periods. Using properties of functions in this class, we prove that in the space of periodic electric potentials in Lloc2(ℝ2) with a given period lattice and identified with L2(K), there exists a second-category set (in the sense of Baire) such that for any electric potential in this set and any homogeneous magnetic field with a rational flow η, the spectrum of the two-dimensional Schrodinger operator is absolutely continuous.

On the Baire Generic Validity of thet-Multifractal Formalism in Besov and Sobolev Spaces

Thet-multifractal formalism is a formula introduced by Jaffard and Mélot in order to deduce thet-spectrum of a functionffrom the knowledge of the(p,t)-oscillation exponent off. Thet-spectrum is the Hausdorff dimension of the set of points wherefhas a given value of pointwiseLtregularity. The(p,t)-oscillation exponent is measured by determining to which oscillation spacesOp,ts(defined in terms of wavelet coefficients)fbelongs. In this paper, we first prove embeddings between oscillation and Besov-Sobolev spaces. We deduce a general lower bound for the(p,t)-oscillation exponent. We then show that this lower bound is actually equality generically, in the sense of Baire’s categories, in a given Sobolev or Besov space. We finally investigate the Baire generic validity of thet-multifractal formalism.

A note on spectrum and quantum dynamics

We show, in the same vein of Simon's Wonderland Theorem, that, typically in Baire's sense, the rates with whom the solutions of the Schrödinger equation escape, in time average, from every finite-dimensional subspace, depend on subsequences of time going to infinite.

Quantum quasiballistic dynamics and thick point spectrum

We obtain dynamical lower bounds for some self-adjoint operators with pure point spectrum in terms of the spacing properties of their eigenvalues. In particular, it is shown that for systems with thick point spectrum, typically in Baire's sense, the dynamics of each initial condition (with respect to some orthonormal bases of the space) presents a quasiballistic behaviour. We present explicit applications to some Schr\"odinger operators.

Equidistribution of minimal hypersurfaces for generic metrics

For almost all Riemannian metrics (in the $$C^\infty $$ Baire sense) on a closed manifold $$M^{n+1}$$ , $$3\le (n+1)\le 7$$ , we prove that there is a sequence of closed, smooth, embedded, connected minimal hypersurfaces that is equidistributed in M. This gives a quantitative version of the main result of Irie et al. (Ann Math 187(3):963–972, 2018), that established density of minimal hypersurfaces for generic metrics. As in Irie et al. (2018), the main tool is the Weyl Law for the Volume Spectrum proven by Liokumovich et al. (Ann Math 187(3):933–961, 2018).

On the manipulability of efficient exchange rules

We identify a large subdomain, D, of quasilinear economies on which any efficient exchange rule will be generically (in the Baire sense) manipulable. For generic economies outside of D, we find rules that are locally non-manipulable. The interior of the set D consists of all economies in which competitive equilibrium would prescribe that all agents consume a positive quantity of money. Since we study quasilinear preferences, this is the domain of primary interest. Our locally non-manipulable rules rely on the existence of traders who are willing to sell all of their cash and absorb the imbalances in the trading of the commodity.

Around Wilson's theorem

We study the series s(n, x) which is the sum for k from 1 to n of the square of the sine of the product x Gamma(k)/k, where x is a variable. By Wilson's theorem we show that the integer part of s(n, x) for x = Pi/2 is the number of primes less or equal to n and we get a similar formula for x a rational multiple of Pi. We show that for almost all x in the Lebesgue measure s(n, x) is equivalent to n/2 when n tends to infinity, while for almost all x in the Baire sense, 1/2 is a limit point of the ratio of s(n, x) to the number of primes less or equal to n.

Density of minimal hypersurfaces for generic metrics

For almost all Riemannian metrics (in the $C^\infty$ Baire sense) on a closed manifold $M^{n+1}$, $3\leq (n+1)\leq 7$, we prove that the union of all closed, smooth, embedded minimal hypersurfaces is dense. This implies there are infinitely many minimal hypersurfaces thus proving a conjecture of Yau (1982) for generic metrics.

Multifractal properties of typical convex functions

We study the singularity (multifractal) spectrum of continuous convex functions defined on $$[0,1]^{d}$$ . Let $$E_f({h}) $$ be the set of points at which f has a pointwise exponent equal to h. We first obtain general upper bounds for the Hausdorff dimension of these sets $$E_f(h)$$ , for all convex functions f and all $$h\ge 0$$ . We prove that for typical/generic (in the sense of Baire) continuous convex functions $$f:[0,1]^{d}\rightarrow { \mathbb {R}}$$ , one has $$\dim E_f(h) =d-2+h$$ for all $$h\in [1,2],$$ and in addition, we obtain that the set $$ E_f({h} )$$ is empty if $$h\in (0,1)\cup (1,+\infty )$$ . Also, when f is typical, the boundary of $$[0,1]^{d}$$ belongs to $$E_{f}({0})$$ .

Topological size of some subsets in certain Calderón–Lozanowskiĭ spaces

For i=1,2,3, let φi be Young functions, (Ω,μ) a (topological) measure space, E an ideal of μ-measurable complex-valued functions defined on Ω and Eφi be the corresponding Calderón–Lozanowskiĭ space. Our aim in this paper is to give, under mild conditions, several results on topological size (in the sense of Baire) of the sets {(f,g)∈Eφ1×Eφ2:|f|⨀|g|∈Eφ3} and {(f,g)∈Eφ1×Eφ2:∃x∈V,(f⨀g)(x) is well defined} where ⨀ denotes the convolution or pointwise product of functions and V a compact neighborhood. Our results sharpen and unify the related results obtained in diverse areas during recent thirty years.

Ergodicity of the Ehrenfest wind–tree model

We consider aperiodic wind–tree models, and show that, for a generic (in the sense of Baire) configuration, the wind–tree dynamics is ergodic in almost every direction.

Minimality of the Ehrenfest wind-tree model

We consider aperiodic wind-tree models, and show that for a generic (in the sense of Baire) configuration the wind-tree dynamics is minimal in almost all directions, and has a dense set of periodic points.

Baire generic results for the anisotropic multifractal formalism

Ben Slimane (Math Proc Camb Philos Soc 124:329–363, 1998) has constructed specific anisotropic selfsimilar functions as counter-examples for the isotropic multifractal formalism. An anisotropic multifractal formalism has been formulated and its validity for anisotropic selfsimilar functions has been proved. In this paper, using Triebel anisotropic wavelet decompositions, we first obtain lower bounds of the anisotropic scaling function and upper bounds of the u-spectrum of singularities valid for all functions. We then investigate the generic validity, in the sense of Baire’s categories, of the anisotropic formalism in some anisotropic functional spaces. We thus extend in the anisotropic setting some results of Jaffard (J Math Pure Appl 79:525–552, 2000, Ann Appl Probab 10:313–329, 2000) and Jaffard and Meyer (Memoirs of the American Mathematical Society, vol. 123, 1996).