Set Of Positive Lebesgue Measure Research Articles

Erdős Similarity Problem via Bi-Lipschitz Embedding

Abstract The Erdős similarity conjecture asserted that an infinite set of real numbers cannot be affinely embedded into every measurable set of positive Lebesgue measure. The problem is still open, in particular for all fast decaying sequences. In this paper, we relax the problem to the bi-Lipschitz embedding and obtain some sharp criteria about the bi-Lipschitz Erdős similarity problem for strictly decreasing sequences.

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.

Positive Measure of Effective Quasi-Periodic Motion Near a Diophantine Torus

It was conjectured by Herman that an analytic Lagrangian Diophantine quasi-periodic torus {mathcal {T}}_0, invariant by a real-analytic Hamiltonian system, is always accumulated by a set of positive Lebesgue measure of other Lagrangian Diophantine quasi-periodic invariant tori. While the conjecture is still open, we will prove the following weaker statement: there exists an open set of positive measure (in fact, the relative measure of the complement is exponentially small) around {mathcal {T}}_0 such that the motion of all initial conditions in this set is “effectively” quasi-periodic in the sense that they are close to being quasi-periodic for an interval of time, which is doubly exponentially long with respect to the inverse of the distance to {mathcal {T}}_0. This open set can be thought of as a neighborhood of a hypothetical invariant set of Lagrangian Diophantine quasi-periodic tori, which may or may not exist.

Dominating Sets in Bergman Spaces on Strongly Pseudoconvex Domains

We obtain local estimates, also called propagation of smallness or Remez-type inequalities, for analytic functions in several variables. Using Carleman estimates, we obtain a three sphere-type inequality, where the outer two spheres can be any sets satisfying a boundary separation property, and the inner sphere can be any set of positive Lebesgue measure. We apply this local result to characterize the dominating sets for Bergman spaces on strongly pseudoconvex domains in terms of a density condition or a testing condition on the reproducing kernels. Our methods also yield a sufficient condition for arbitrary domains and lower-dimensional sets.

Chaotic Dynamics of Spatially Homogeneous Spacetimes

In the 1970’s, Belinskii, Khalatnikov and Lifshitz have proposed a conjectural description of the asymptotic geometry of cosmological models in the vicinity of their initial singularity. In particular, it is believed that the asymptotic geometry of generic spatially homogeneous spacetimes should display an oscillatory chaotic behaviour modeled on a discrete map’s dynamics (the so-called Kasner map). We prove that this conjecture holds true, if not for generic spacetimes, at least for a positive Lebesgue measure set of spacetimes. In the context of spatially homogeneous spacetimes, the Einstein field equations can be reduced to a system of differential equations on a finite dimensional phase space: the Wainwright–Hsu equations. The dynamics of these equations encodes the evolution of the geometry of spacelike slices in spatially homogeneous spacetimes. Our proof is based on the non-uniform hyperbolicity of the Wainwright–Hsu equations. Indeed, we consider the return map of the solutions of these equations on a transverse section and prove that it is a non-uniformly hyperbolic map with singularities. This allows us to construct some local stable manifolds à la Pesin for this map and to prove that the union of the orbits starting in these local stable manifolds cover a positive Lebesgue measure set in the phase space. The chaotic oscillatory behaviour of the corresponding spacetimes follows.

Some properties of Vitali sets

Abstract If V 1 , … , V n {V_{1},\dots,V_{n}} are translations of a Vitali set by rational numbers, then we prove that ⋃ i = 1 n V i {\bigcup_{i=1}^{n}V_{i}} contains no measurable subset of positive measure. This provides a decomposition of ℝ {{\mathbb{R}}} as a countable union of disjoint sets, any finite union of which has Lebesgue inner measure zero. As a consequence, we present a function δ : ℝ → ( 0 , + ∞ ) {\delta:{\mathbb{R}}\rightarrow(0,+\infty)} for which there is no measurable function f : ℝ → ℝ {f:{\mathbb{R}}\rightarrow{\mathbb{R}}} satisfying 0 &lt; f ≤ δ {0&lt;f\leq\delta} , on any measurable set of positive Lebesgue measure.

Non-uniqueness of integral curves for autonomous Hamiltonian vector fields

In this work we prove the existence of an autonomous Hamiltonian vector field in $W^{1,r}(\mathbb T^d; \mathbb R^d)$ with $r < d-1$ and $d \geq 4$ for which the associated transport equation has non-unique positive solutions. As a consequence of Ambrosio's superposition principle [2], we show that this vector field has non-unique integral curves with a positive Lebesgue measure set of initial data and moreover, we show that the Hamiltonian is not constant along these integral curves.

SEPARATELY-ANALYTIC FUNCTION WITH SINGULARITIES SET OF POSITIVE LEBESGUE MEASURE

In this work Shimoda’s theorem on separately-analytic functions is discussed and there is constructed an example of separately-analytic function with singularities set of positive Lebesgue measure. Proved that, separately-analytic functions satisfying Shimoda’s theorem on can be represented by Hartogs’ series with holomorphic coefficients out of some nowhere dense compact subset of .

Abundance of Observable Lyapunov Irregular Sets

Lyapunov exponent is widely used in natural science to find chaotic signal, but its existence is seldom discussed. In the present paper, we consider the problem of whether the set of points at which Lyapunov exponent fails to exist, called the Lyapunov irregular set, has positive Lebesgue measure. The only known example with the Lyapunov irregular set of positive Lebesgue measure is a figure-8 attractor by the work of Ott and Yorke (Phys Rev 78, 056203, 2008), whose key mechanism (homoclinic loop) is easy to be broken by small perturbations. In this paper, we show that surface diffeomorphisms with a robust homoclinic tangency given by Colli and Vargas (Ergod Theory Dyn Syst 21, 1657–1681, 2001), as well as other several known nonhyperbolic dynamics, have the Lyapunov irregular set of positive Lebesgue measure. We can construct such positive Lebesgue measure sets both as the time averages exist and do not exist on it.

Propagation of smallness and control for heat equations

In this note we investigate propagation of smallness properties for solutions to heat equations. We consider spectral projector estimates for the Laplace operator with Dirichlet or Neumann boundary conditions on a Riemanian manifold with or without boundary. We show that using the new approach for the propagation of smallness of Logunov and Malinnikova (2018) allows one to extend the spectral projector type estimates of Jerison and Lebeau (1999) from localisation on open sets to localization on arbitrary sets of non-zero Lebesgue measure; we can actually go beyond and consider sets of non-vanishing d- \delta ( \delta &gt;0 small enough) Hausdorff measure. We show that these new spectral projector estimates allow one to extend Logunov–Malinnikova’s propagation of smallness results to solutions to heat equations. Finally, we apply these results to the null controllability of heat equations with controls localized on sets of positive Lebesgue measure. The main novelty here is that we can drop the constant coefficient assumptions of Apraiz et al. (2013, 2014) on the Laplace operator (or the analyticity assumption of Escauriaza et al. (2017) and Lebeau and Moyano (2019)) and deal with Lipschitz coefficients. Another important novelty is that we get the first (non-one-dimensional) exact controllability results with controls supported on measure zero sets.

Lebesgue measure of Feigenbaum Julia sets

We construct Feigenbaum quadratic-like maps with a Julia set of positive Lebesgue measure. Indeed, in the quadratic family $P_c: z \mapsto z^2+c$ the corresponding set of parameters $c$ is shown to have positive Hausdorff dimension. Our examples include renormalization fixed points, and the corresponding quadratic polynomials in their stable manifold are the first known rational maps for which the hyperbolic dimension is different from the Hausdorff dimension of the Julia set.

Trees of Dot Products in Thin Subsets of Rd

A. Iosevich and K. Taylor showed that compact subsets of Rd with Hausdorff dimension greater than (d+1)/2 contain trees with gaps in an open interval. Under the same dimensional threshold, we prove the analogous result where distance is replaced by the dot product. We additionally show that the gaps of embedded trees of dot products are prevalentin a set of positive Lebesgue measure, and for Ahlfors-David regular sets, the number of treeswith given gaps agrees with the regular value theorem.

Lyapunov exponents for random maps

&lt;p style='text-indent:20px;'&gt;It has been recently realized that for abundant dynamical systems on a compact manifold, the set of points for which Lyapunov exponents fail to exist, called the Lyapunov irregular set, has positive Lebesgue measure. In the present paper, we show that under any physical noise, the Lyapunov irregular set has zero Lebesgue measure and the number of such Lyapunov exponents is finite. This result is a Lyapunov exponent version of Araújo's theorem on the existence and finitude of time averages. Furthermore, we numerically compute the Lyapunov exponents for a surface flow with an attracting heteroclinic connection, which enjoys the Lyapunov irregular set of positive Lebesgue measure, under a physical noise. This paper also contains the proof of the disappearance of Lyapunov irregular behavior on a positive Lebesgue measure set for a surface flow with an attracting homoclinic/heteroclinic connection under a non-physical noise.&lt;/p&gt;

Analogues of Beurling's theorem for some integral transforms

A famous result of Beurling says that if the Fourier transform of a non-zero integrable function on the real line has certain exponential decay, then the function cannot vanish on a set of positive Lebesgue measure. We prove an analogue of Beurling's theorem for Hankel transform and some several variable analogues of Beurling's theorem for Fourier transform as a consequence of similar results for Dunkl transform. We also prove an analogue of Beurling's theorem for spectral projections associated to the Dunkl-Laplacian.

Coexistence of absolutely continuous and pure point spectrum for kicked quasiperiodic potentials

We introduce a class of real analytic “peaky” potentials for which the corresponding quasiperiodic 1D-Schrödinger operators exhibit, for quasiperiodic frequencies in a set of positive Lebesgue measure, both absolutely continuous and pure point spectrum.

Sharp cohomological bound for uniformly quasiregularly elliptic manifolds

We show that if a compact, connected, and oriented $n$-manifold $M$ without boundary admits a non-constant non-injective uniformly quasiregular self-map, then the dimension of the real singular cohomology ring $H^*(M;\Bbb{R})$ of $M$ is bounded from above by $2^n$. This is a positive answer to a dynamical counterpart of the Bonk-Heinonen conjecture on the cohomology bound for quasiregularly elliptic manifolds. The proof is based on an intermediary result that, if $M$ is not a rational homology sphere, then each such uniformly quasiregular self-map on $M$ has a Julia set of positive Lebesgue measure.

Abundance of Strange Attractors Near an Attracting Periodically Perturbed Network

We study the dynamics of the periodically forced May--Leonard system. We extend previous results on the field, and we identify different dynamical regimes depending on the strength of attraction $\delta$ of the network and the frequency $\omega$ of the periodic forcing. We focus our attention in the case $\delta\gg1$ and $\omega \approx 0$, where we show that, for a positive Lebesgue measure set of parameters (amplitude of the periodic forcing), the dynamics are dominated by strange attractors with fully stochastic properties, supporting Sinai--Ruelle--Bowen (SRB) measures. The proof is performed by using the Wang and Young's theory of rank-one strange attractors. This work ends the discussion about the existence of observable and sustainable chaos in this scenario. We also identify some bifurcations occurring in the transition from an attracting two-torus to rank-one strange attractors, whose existence has been suggested by numerical simulations.

The norm of the characteristic function of a set in the John‐Nirenberg space of exponent p

We find the concrete value of for any measurable set of positive and finite Lebesgue measure, where stands for the John‐Nirenberg space of exponent . In the case we show that for any interval with and any where

Observability Inequalities on Measurable Sets for the Stokes System and Applications

In this paper, we establish spectral inequalities on measurable sets of positive Lebesgue measure for the Stokes operator, as well as observability inequalities on space-time measurable sets of positive measure for nonstationary Stokes system. The latter extends the result established recently by Wang and Zhang [SIAM J. Control Optim., 55 (2017), pp. 1862--1886] to the case of observations from subsets of positive measure in both time and space variables. Furthermore, we present their applications in the shape optimization problem, as well as the time optimal control problem for the Stokes system. In particular, we give a positive answer to an open question raised by Privat, Trélat, and Zuazua [Arch. Rational Mech. Anal., 216 (2015), pp. 921--981].

On exponential decay of the variance of BLUE for the mean of a stationary sequence

Abstract In this paper, we obtain necessary as well as sufficient conditions for exponential rate of decrease of the variance of the best linear unbiased estimator (BLUE) for the unknown mean of a stationary sequence possessing a spectral density. In particular, we show that a necessary condition for variance of BLUE to decrease to zero exponentially is that the spectral density vanishes on a set of positive Lebesgue measure in any vicinity of zero.