Sense Of Baire Research Articles

Baire Typical Results for Mixed Hölder Spectra on Product of Continuous Besov or Oscillation Spaces

In this paper, we show that the mixed Holder spectra of finitely many functions can be recovered from a Legendre transform of a concave scaling function based on simultaneous wavelet leaders. We first prove that this Legendre transform yields an upper bound valid for uniform Holder functions. We then study the typical optimality (in the sense of Baire’s category) of this upper bound in a product of continuous Besov or oscillation spaces.

CARATHEODORY DIFFERENTIAL EQUATIONS ON CONES

In this paper we prove that almost all, in Baire sense, differential equations with Scorza Dragoni right-hand side, defined on closed convex cone of a Banach space, have unique solution. This solution depends continuously on the right-hand side and on the initial condition. The results are applied to fuzzy differential equations and to differential inclusions.

Non-normal numbers in dynamical systems fulfilling the specification property

In the present paper we want to focus on the dichotomy of the non-normal numbers -- on the one hand they are a set of measure zero and on the other hand they are residual -- for dynamical system fulfilling the specification property. These dynamical systems are motivated by $\beta$-expansions. We consider the limiting frequencies of digits in the words of the languagse arising from these dynamical systems, and show that not only a typical $x$ in the sense of Baire is non-normal, but also its Cesàro variants diverge.

Dimension and measure for typical random fractals

AbstractWe define a random iterated function system (RIFS) to be a finite set of (deterministic) iterated function systems (IFSs) acting on the same metric space. For a given RIFS, there exists a continuum of random attractors corresponding to each sequence of deterministic IFSs. Much work has been done on computing the ‘almost sure’ dimensions of these random attractors. Here we compute the typical dimensions (in the sense of Baire) and observe that our results are in stark contrast to those obtained using the probabilistic approach. Furthermore, we examine the typical Hausdorff and packing measures of the random attractors and give examples to illustrate some of the strange phenomena that can occur. The only restriction we impose on the maps is that they are bi-Lipschitz and we obtain our dimension results without assuming any separation conditions.

No-where averagely differentiable functions: Baire category and the Takagi function

A classical and well-known result due to Banach and Mazurkiewicz says that a typical (in the sense of Baire) continuous function on the unit interval is no-where differentiable. In this paper we prove that a typical (in the sense of Baire) continuous function \(f\) on [0, 1] is spectacularly more irregular than suggested by Banach and Mazurkiewicz’s result. Namely, not only is the difference quotient \(\frac{f(x+h)-f(x)}{h}\) divergent as \(h\rightarrow 0\) for all \(x\), but the function \(h\rightarrow \frac{f(x+h)-f(x)}{h}\) diverges so badly as \(h\rightarrow 0\), that it remains divergent even after being “smoothened out” using iterated Cesaro averages of arbitrary high order. More precisely, we introduce the notion of higher order average differentiability based on iterated Cesaro averages and prove that not only is a typical (in the sense of Baire) continuous function on [0, 1] no-where differentiable, but it is even no-where averagely differentiable of any order. We also show that the no-where differentiable Takagi function is, in fact, no-where averagely differentiable.

Free subsemigroups in automorphism group of a polynomial ring of two variables over number fields

Sufficient conditions under which the semigroup generated by two automorphism of the polynomial ring in two variables over a number field $P$ will be free are given. The set of free finitely generated subsemigroups of the triangle automorphism subgroup in $AutP[x; y]$ and the set of non-free finitely generated of semigroups of this groups are compared in categorial sense of Baire.

Nonmonotone equations with large almost periodic forcing terms

We consider the scalar differential equation u˙=f(u)+ch(t) where f(u) is a jumping nonlinearity and h(t) is an almost periodic function, while c is a real parameter deciding the size of the forcing term. The main result is that, if h(t) does not vanish too much in some suitable sense, then the equation admits a (unique) almost periodic solution for large values of the parameter c. The class of the h(t)ʼs to which the result applies is studied in detail: it includes all the nontrivial trigonometric polynomials and is generic in the Baire sense.

Generic Properties of Continuous Differential Inclusions and the Tonelli Method of Approximate Solutions

We study Cauchy problems for differential inclusions in Banach spaces and show that most such problems (in the sense of Baire’s categories) have solutions. We consider separately the cases where the point images of the right-hand side are compact and convex, and where they are merely bounded, closed and convex.

On the box dimensions of graphs of typical continuous functions

Let X⊆R be a bounded set; we emphasize that we are not assuming that X is compact or Borel. We prove that for a typical (in the sense of Baire) uniformly continuous function f on X, the lower box dimension of the graph of f is as small as possible and the upper box dimension of the graph of f is as big as possible. We also prove a local version of this result. Namely, we prove that for a typical uniformly continuous function f on X, the lower local box dimension of the graph of f at all points x∈X is as small as possible and the upper local box dimension of the graph of f at all points x∈X is as big as possible.

The multifractal box dimensions of typical measures

We compute the typical (in the sense of Baire's category theorem) multifractal box dimensions of measures on a compact subset of $\mathbb R^d$. Our results are new even in the context of box dimensions of measures.

Typical multifractal box dimensions of measures

We study the typical behaviour (in the sense of Baire's category) of the multifractal box dimensions of measures on $\mathbb R^{d}$. We prove that in many cases a typical measure $\mu$ is as irregular as possible, i.e. the lower multifractal box dimensio

Generic properties of multifunctions: Application to differential inclusions

We prove that almost all in Baire sense Caratheodory multifunctions in finite dimensional space are Kamke continuous. Further the main properties of differential inclusions with Kamke and one sided Kamke right-hand sides are studied.As a corollary we prove that for almost all optimal control problems, the relaxation and relaxation stability properties hold.

Typical Recurrence for the Ehrenfest Wind-Tree Model

We show that the typical wind-tree model, in the sense of Baire, is recurrent and has a dense set of periodic orbits. The recurrence result also holds for the Lorentz gas: the typical Lorentz gas, in the sense of Baire, is recurrent. These Lorentz gases need not be of finite horizon!

Typical behavior of mixed [formula omitted]-dimensions

Previously, L. Olsen had studied the typical behavior (in the sense of Baire’s category) of the upper and lower L q -dimensions of a single measure. In this paper, we will focus on the typical behavior of the upper and lower mixed L q -dimensions of finitely many measures.

SINGULAR SPECTRUM FOR RADIAL TREES

We prove several results showing that absolutely continuous spectrum for the Laplacian on radial trees is a rare event. In particular, we show that metric trees with unbounded edges have purely singular spectrum and that, generically (in the sense of Baire), radial trees have purely singular continuous spectrum.

Baire generic histograms of wavelet coefficients and large deviation formalism in Besov and Sobolev spaces

Histograms of wavelet coefficients are expressed in terms of the wavelet profile and the wavelet density. The large deviation multifractal formalism states that if a function f has a minimal uniform Hölder regularity then its Hölder spectrum is equal to the wavelet density. The purpose of this paper is twofold. Firstly, we compute generically (in the sense of Baire's categories) these histograms in Besov B p s , q ( T ) and L p , s ( T ) spaces, where T is the torus R d / Z d (resp. in the Baire's vector space V = ⋂ ε > 0 , p > 0 B p s ( 1 p ) − ε p , p where s : q ↦ s ( q ) is a C 1 and concave function on R + satisfying 0 ⩽ s ′ ⩽ d and s ( 0 ) > 0 ). Secondly, as an application, we deduce some extra generic properties for the histograms in these spaces, and study the generic validity of the large deviation multifractal formalism in Besov and L p , s spaces for s > d / p (resp. in the above space V).

Nowhere monotone functions and microscopic sets

We investigate how large a set can be on which a continuous nowhere monotone function is one-to-one. We consider the σ-ideal of microscopic sets, which is situated between the countable sets and the sets of Hausdorff dimension zero and prove that the typical function in C[0, 1] (in the sense of Baire) is nowhere monotone and one-to-one except on some microscopic set. We also give an example of a continuous nowhere monotone function of bounded variation on [0, 1], which is one-to-one except on some microscopic set, so it is not a typical function.

Typical upper [formula omitted]-dimensions of measures for [formula omitted

For a probability measure μ on a subset of R d , the lower and upper L q -dimensions of order q ∈ R are defined by D ̲ μ ( q ) = lim inf r ↘ 0 log ∫ μ ( B ( x , r ) ) q − 1 d μ ( x ) − log r , D ¯ μ ( q ) = lim sup r ↘ 0 log ∫ μ ( B ( x , r ) ) q − 1 d μ ( x ) − log r . In previous work we studied the typical behaviour (in the sense of Baire's category) of the L q -dimensions D ̲ μ ( q ) and D ¯ μ ( q ) for q ⩾ 1 . In the present work we study the typical behaviour (in the sense of Baire's category) of the upper L q -dimensions D ¯ μ ( q ) for q ∈ [ 0 , 1 ] .

Nonlinear Hyperbolic Systems: Nondegenerate Flux, Inner Speed Variation, and Graph Solutions

We study the Cauchy problem for general nonlinear strictly hyperbolic systems of partial differential equations in one space variable. First, we re-visit the construction of the solution to the Riemann problem and introduce the notion of a nondegenerate (ND) system. This is the optimal condition guaranteeing, as we show it, that the Riemann problem can be solved with finitely many waves only; we establish that the ND condition is generic in the sense of Baire (for the Whitney topology), so that any system can be approached by a ND system. Second, we introduce the concept of inner speed variation and we derive new interaction estimates on wave speeds. Third, we design a wave front tracking scheme and establish its strong convergence to the entropy solution of the Cauchy problem; this provides a new existence proof as well as an approximation algorithm. As an application, we investigate the time regularity of the graph solutions (X,U) introduced by LeFloch, and propose a geometric version of our scheme; in turn, the spatial component X of a graph solution can be chosen to be continuous in both time and space, while its component U is continuous in space and has bounded variation in time.

Typical Rényi dimensions of measures. The cases: [formula omitted] and [formula omitted

We study the typical behaviour (in the sense of Baire's category) of the q-Rényi dimensions D ̲ μ ( q ) and D ¯ μ ( q ) of a probability measure μ on R d for q ∈ [ − ∞ , ∞ ] . Previously we found the q-Rényi dimensions D ̲ μ ( q ) and D ¯ μ ( q ) of a typical measure for q ∈ ( 0 , ∞ ) . In this paper we determine the q-Rényi dimensions D ̲ μ ( q ) and D ¯ μ ( q ) of a typical measure for q = 1 and for q = ∞ . In particular, we prove that a typical measure μ is as irregular as possible: for q = ∞ , the lower Rényi dimension D ̲ μ ( q ) attains the smallest possible value, and for q = 1 and q = ∞ the upper Rényi dimension D ¯ μ ( q ) attains the largest possible value.