Lebesgue Measure Research Articles

Cramér distance and discretisations of circle expanding maps I: theory

This paper is aimed to study the ergodic short-term behaviour of discretisations of circle expanding maps. More precisely, we prove some asymptotics of the distance between the tth iterate of Lebesgue measure by the dynamics f and the tth iterate of the uniform measure on the grid of order N by the discretisation on this grid, when t is fixed and the order N goes to infinity. This is done under some explicit genericity hypotheses on the dynamics, and the distance between measures is measured by the mean of Cramér distance. The proof is based on a study of the corresponding linearised problem, where the problem is translated into terms of equirepartition on tori of dimension exponential in t. A numerical study associated to this work is presented in Guihéneuf and Monge (2022 arXiv:2206.08000).

A modified Christoffel function and its asymptotic properties

We introduce a certain variant (or regularization) Λ̃nμ of the standard Christoffel function Λnμ associated with a measure μ on a compact set Ω⊂Rd. Its reciprocal is now a sum-of-squares polynomial in the variables (x,ɛ), ɛ>0. It shares the same dichotomy property of the standard Christoffel function, that is, the growth with n of its inverse is at most polynomial inside and exponential outside the support of the measure. Its distinguishing and crucial feature states that for fixed ɛ>0, and under weak assumptions, limn→∞ɛ−dΛ̃nμ(ξ,ɛ)=f(ζɛ) where f (assumed to be continuous) is the unknown density of μ w.r.t. Lebesgue measure on Ω, and ζɛ∈B∞(ξ,ɛ) (and so f(ζɛ)≈f(ξ) when ɛ>0 is small). This is in contrast with the standard Christoffel function where if limn→∞ndΛnμ(ξ) exists, it is of the form f(ξ)/ωE(ξ) where ωE is the density of the equilibrium measure of Ω, usually unknown. At last but not least, the additional computational burden (when compared to computing Λnμ) is just integrating symbolically the monomial basis (xα)α∈Nnd on the box {x:‖x−ξ‖∞<ɛ/2}, so that 1/Λ̃nμ is obtained as an explicit polynomial of (ξ,ɛ).

Some results on the comparison principle for the weighted log canonical thresholds

The purpose of this paper is to establish some results on the comparison principle for the weighted log canonical thresholds of plurisubharmonic functions in case the weight is a measure of the form [Formula: see text] and [Formula: see text] with [Formula: see text], where [Formula: see text] is the Lebesgue measure in [Formula: see text].

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} .

Coexistence Phenomena in the Hénon Family

We study the classical Hénon family f_{a,b}:(x,y)mapsto (1-ax^2+y,bx), 0<a<2, 0<b<1, and prove that given an integer kge 1, there is a set of parameters E_k of positive two-dimensional Lebesgue measure so that f_{a,b}, for (a,b)in E_k, has at least k attractive periodic orbits and one strange attractor of the type studied in Benedicks and Carleson (Ann Math (2) 133(1):73–169, 1991). A corresponding statement also holds for the Hénon-like families of Mora and Viana (Acta Math 171:1–71, 1993), and we use the techniques of Mora and Viana (1993) to study homoclinic unfoldings also in the case of the original Hénon maps. The final main result of the paper is the existence, within the classical Hénon family, of a positive Lebesgue measure set of parameters whose corresponding maps have two coexisting strange attractors.

Densely branching trees as models for Hénon-like and Lozi-like attractors

Inspired by a recent work of Crovisier and Pujals on mildly dissipative diffeomorphisms of the plane, we show that Hénon-like and Lozi-like maps on their strange attractors are conjugate to natural extensions (a.k.a. shift homeomorphisms on inverse limits) of maps on metric trees with dense set of branch points. In consequence, these trees very well approximate the topology of the attractors, and the maps on them give good models of the dynamics. To the best of our knowledge, these are the first examples of canonical two-parameter families of attractors in the plane for which one is guaranteed such a 1-dimensional locally connected model tying together topology and dynamics of these attractors. For the Hénon maps this applies to a positive Lebesgue measure parameter set generalizing the Benedicks-Carleson parameters, the Wang-Young parameter set, and sheds more light onto the result of Barge from 1987, who showed that there exist parameter values for which Hénon maps on their attractors are not natural extensions of any maps on branched 1-manifolds. For the Lozi maps the result applies to an open set of parameters given by Misiurewicz in 1980. Our result can be seen as a generalization to the non-uniformly hyperbolic world of a classical result of Williams from 1967. We also show that no simpler 1-dimensional models exist.

PDE for the joint law of the pair of a continuous diffusion and its running maximum

AbstractLet X be a d-dimensional diffusion and M the running supremum of its first component. In this paper, we show that for any $t&gt;0,$ the density (with respect to the $(d+1)$ -dimensional Lebesgue measure) of the pair $\big(M_t,X_t\big)$ is a weak solution of a Fokker–Planck partial differential equation on the closed set $\big\{(m,x)\in \mathbb{R}^{d+1},\,{m\geq x^1}\big\},$ using an integral expansion of this density.

On the boundedness and norm of certain generalized Hilbert operators in ℓp$\ell ^p$

AbstractThe Hilbert matrix has been extensively studied in the previous literature. In this paper, we look at generalized Hilbert operators arising from measures on the interval [0,1], such that the Hilbert matrix is obtained by the Lebesgue measure. We provide a necessary and sufficient condition for these operators to be bounded in and calculate their norm. Interestingly, we obtain complete results in this direction for both .

Generalised Hausdorff measure of sets of Dirichlet non-improvable matrices in higher dimensions

Let psi :mathbb {R}_{+}rightarrow mathbb {R}_{+} be a non-increasing function. A pair (A,{textbf{b}}), where A is a real mtimes n matrix and {textbf{b}}in mathbb {R}^{m}, is said to be psi -Dirichlet improvable, if the system ‖Aq+b-p‖m<ψ(T),‖q‖n<T\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\Vert A{\ extbf{q}} +{\ extbf{b}}-{\ extbf{p}}\\Vert ^m<\\psi (T), \\quad \\Vert {\ extbf{q}}\\Vert ^n<T \\end{aligned}$$\\end{document}is solvable in {textbf{p}}in mathbb {Z}^{m},{textbf{q}}in mathbb {Z}^{n} for all sufficiently large T where Vert cdot Vert denotes the supremum norm. For psi -Dirichlet non-improvable sets, Kleinbock–Wadleigh (2019) proved the Lebesgue measure criterion whereas Kim–Kim (2022) established the Hausdorff measure results. In this paper we obtain the generalised Hausdorff f-measure version of Kim–Kim (2022) results for psi -Dirichlet non-improvable sets.

A generalization of van der Corput's difference theorem with applications to recurrence and multiple ergodic averages

We prove a generalization of van der Corput's difference theorem for sequences of vectors in a Hilbert space. This generalization is obtained by establishing a connection between sequences of vectors in the first Hilbert space with a vector in a new Hilbert space whose spectral type with respect to a certain unitary operator is absolutely continuous with respect to the Lebesgue measure. We use this generalization to obtain applications regarding recurrence and multiple ergodic averages when we have measure preserving automorphisms T and S that do not necessarily commute, but T has a maximal spectral type that is mutually singular with the Lebesgue measure.

Strong Asymptotics of Planar Orthogonal Polynomials: Gaussian Weight Perturbed by Finite Number of Point Charges

AbstractWe consider the orthogonal polynomial pn(z) with respect to the planar measure supported on the whole complex plane where dA is the Lebesgue measure of the plane, N is a positive constant, {c1, …, cν} are nonzero real numbers greater than −1 and are distinct points inside the unit disk. In the scaling limit when n/N = 1 and n → ∞ we obtain the strong asymptotics of the polynomial pn(z). We show that the support of the roots converges to what we call the “multiple Szegő curve,” a certain connected curve having ν + 1 components in its complement. We apply the nonlinear steepest descent method [9,10] on the matrix Riemann‐Hilbert problem of size (ν + 1) × (ν + 1) posed in [22]. © 2023 Wiley Periodicals, LLC.

Overlapping Iterated Function Systems from the Perspective of Metric Number Theory

In this paper we develop a new approach for studying overlapping iterated function systems. This approach is inspired by a famous result due to Khintchine from Diophantine approximation which shows that for a family of limsup sets, their Lebesgue measure is determined by the convergence or divergence of naturally occurring volume sums. For many parameterised families of overlapping iterated function systems, we prove that a typical member will exhibit similar Khintchine like behaviour. Families of iterated function systems that our results apply to include those arising from Bernoulli convolutions, the{0,1,3}\{0,1,3\}problem, and affine contractions with varying translation parameter. As a by-product of our analysis we obtain new proofs of some well known results due to Solomyak on the absolute continuity of Bernoulli convolutions, and when the attractor in the{0,1,3}\{0,1,3\}problem has positive Lebesgue measure.For eacht∈[0,1]t\in [0,1]we letΦt\Phi _tbe the iterated function system given byΦt≔{ϕ1(x)=x2,ϕ2(x)=x+12,ϕ3(x)=x+t2,ϕ4(x)=x+1+t2}.\begin{equation*} \Phi _{t}≔\Big \{\phi _1(x)=\frac {x}{2},\phi _2(x)=\frac {x+1}{2},\phi _3(x)=\frac {x+t}{2},\phi _{4}(x)=\frac {x+1+t}{2}\Big \}. \end{equation*}We prove that eitherΦt\Phi _tcontains an exact overlap, or we observe Khintchine like behaviour. Our analysis shows that by studying the metric properties of limsup sets, we can distinguish between the overlapping behaviour of iterated function systems in a way that is not available to us by simply studying properties of self-similar measures.Last of all, we introduce a property of an iterated function system that we call being consistently separated with respect to a measure. We prove that this property implies that the pushforward of the measure is absolutely continuous. We include several explicit examples of consistently separated iterated function systems.

On Landau-Kolmogorov type inequalities for charges and their applications

In this article we prove sharp Landau-Kolmogorov type inequalities on a class of charges defined on Lebesgue measurable subsets of a cone in $\mathbb{R}^d$, $d\geqslant 1$, that are absolutely continuous with respect to the Lebesgue measure. In addition we solve the Stechkin problem of approximation of the Radon-Nikodym derivative of such charges by bounded operators and two related problems. As an application, we also solve these extremal problems on classes of essentially bounded functions $f$ such that their distributional partial derivative $\frac{\partial ^d f}{\partial x_1\ldots\partial x_d}$ belongs to the Sobolev space $W^{1,\infty}$.

Thin spectra and singular continuous spectral measures for limit‐periodic Jacobi matrices

AbstractThis paper investigates the spectral properties of Jacobi matrices with limit‐periodic coefficients. We show that generically the spectrum is a Cantor set of zero Lebesgue measure, and the spectral measures are purely singular continuous. For a dense set of limit‐periodic Jacobi matrices, we show that the spectrum is a Cantor set of zero lower box counting dimension while still retaining the singular continuity of the spectral type. We also show how results of this nature can be established by fixing the off‐diagonal coefficients and varying only the diagonal coefficients, and, in a more restricted version, by fixing the diagonal coefficients to be zero and varying only the off‐diagonal coefficients. We apply these results to produce examples of weighted Laplacians on the multidimensional integer lattice having purely singular continuous spectral type and zero‐dimensional spectrum.

Различие мер Хаара цилиндров в теореме Дирихле для поля p-адических чисел

The Dirichlet box principle gives surprisingly accurate results in problems of approximation of real numbers by rational numbers, transcendental numbers by real algebraic numbers. Every polynomial taking small values at a given point $x$ also takes small values in its neighborhood. A problem of studying such neighborhoods and obtaining possible Lebesgue measure values arises frequently. In this paper we solve the problem in the p-adic case using recent results of the metric theory of Diophantine approximations.

Stochastic Fokker–Planck Equations for Conditional McKean–Vlasov Jump Diffusions and Applications to Optimal Control

The purpose of this paper is to study optimal control of conditional McKean–Vlasov (mean-field) stochastic differential equations with jumps (conditional McKean–Vlasov jump diffusions, for short). To this end, we first prove a stochastic Fokker–Planck equation for the conditional law of the solution of such equations. Combining this equation with the original state equation, we obtain a Markovian system for the state and its conditional law. Furthermore, we apply this to formulate a Hamilton–Jacobi–Bellman equation for the optimal control of conditional McKean–Vlasov jump diffusions. Then we study the situation when the law is absolutely continuous with respect to Lebesgue measure. In that case the Fokker–Planck equation reduces to a stochastic partial differential equation for the Radon–Nikodym derivative of the conditional law. Finally we apply these results to solve explicitly the linear-quadratic optimal control problem of conditional stochastic McKean–Vlasov jump diffusions, and optimal consumption from a cash flow.

Measure theoretic laws for limsup sets defined by rectangles

In this paper, we present a general principle for the Lebesgue measure theory of limsup sets defined by rectangles under the hypothesis of ubiquity for rectangles. Applications are given for simultaneous Diophantine approximation, Khintchine-Groshev theorem for rectangles and higher dimensional shrinking target problems.

Using Bernoulli maps to accelerate mixing of a random walk on the torus

We study the mixing time of a random walk on the torus, alternated with a Lebesgue measure preserving Bernoulli map. Without the Bernoulli map, the mixing time of the random walk alone is O ( 1 / ε 2 ) O(1/\varepsilon ^2) , where ε \varepsilon is the step size. Our main results show that for a class of Bernoulli maps, when the random walk is alternated with the Bernoulli map φ \varphi the mixing time becomes O ( | ln ⁡ ε | ) O(\lvert \ln \varepsilon \rvert ) . We also study the dissipation time of this process, and obtain O ( | ln ⁡ ε | ) O(\lvert \ln \varepsilon \rvert ) upper and lower bounds with explicit constants.

On the size of A+λA for algebraic λ

For a finite set $A\subset \mathbb{R}$ and real $\lambda$, let $A+\lambda A:=\{a+\lambda b :\, a,b\in A\}$. Combining a structural theorem of Freiman on sets with small doubling constants together with a discrete analogue of Pr\'ekopa--Leindler inequality we prove a lower bound $|A+\sqrt{2} A|\geq (1+\sqrt{2})^2|A|-O({|A|}^{1-\varepsilon})$ which is essentially tight. We also formulate a conjecture about the value of $\liminf |A+\lambda A|/|A|$ for an arbitrary algebraic $\lambda$. Finally, we prove a tight lower bound on the Lebesgue measure of $K+\mathcal{T} K$ for a given linear operator $\mathcal{T}\in \operatorname{End}(\mathbb{R}^d)$ and a compact set $K\subset \mathbb{R}^d$ with fixed measure. This continuous result supports the conjecture and yields an upper bound in it.

Dynamics of Lebesgue Quadratic Stochastic Operator with Nonnegative Integers Parameters Generated by 2-PartitionSiti Nurlaili Karima, Nur Zatul Akmar Hamzahb, Nasir Ganikhodajevc, Muhammad Azrin Ahmadd, Norarida Abd RhanieaFaculty of Information and Communication Technology, Universiti Tunku Abdul Rahman, Kampar, Perak, Malaysia; bDepartment of Computational and Theoretical Sciences, International Islamic University Malaysia, Kuantan, Pahang, Malaysia; cInstitute of Mathematics, Uzbekistan Academy of Sciences, Tashkent,

Dynamics of Lebesgue Quadratic Stochastic Operator with Nonnegative Integers Parameters Generated by 2-PartitionSiti Nurlaili Karima, Nur Zatul Akmar Hamzahb, Nasir Ganikhodajevc, Muhammad Azrin Ahmadd, Norarida Abd RhanieaFaculty of Information and Communication Technology, Universiti Tunku Abdul Rahman, Kampar, Perak, Malaysia; bDepartment of Computational and Theoretical Sciences, International Islamic University Malaysia, Kuantan, Pahang, Malaysia; cInstitute of Mathematics, Uzbekistan Academy of Sciences, Tashkent,