Set Of Full Lebesgue Measure Research Articles

Quasisymmetric orbit-flexibility of multicritical circle maps

AbstractTwo given orbits of a minimal circle homeomorphism f are said to be geometrically equivalent if there exists a quasisymmetric circle homeomorphism identifying both orbits and commuting with f. By a well-known theorem due to Herman and Yoccoz, if f is a smooth diffeomorphism with Diophantine rotation number, then any two orbits are geometrically equivalent. It follows from the a priori bounds of Herman and Świątek, that the same holds if f is a critical circle map with rotation number of bounded type. By contrast, we prove in the present paper that if f is a critical circle map whose rotation number belongs to a certain full Lebesgue measure set in $(0,1)$ , then the number of equivalence classes is uncountable (Theorem 1.1). The proof of this result relies on the ergodicity of a two-dimensional skew product over the Gauss map. As a by-product of our techniques, we construct topological conjugacies between multicritical circle maps which are not quasisymmetric, and we show that this phenomenon is abundant, both from the topological and measure-theoretical viewpoints (Theorems 1.6 and 1.8).

A criterion for the triviality of the centralizer for vector fields and applications

In this paper we establish a criterion for the triviality of the C1-centralizer for vector fields and flows. In particular we deduce the triviality of the centralizer at homoclinic classes of Cr vector fields (r≥1). Furthermore, we show that the set of flows whose C1-centralizer is trivial include: (i) C1-generic volume preserving flows, (ii) C2-generic Hamiltonian flows on a generic and full Lebesgue measure set of energy levels, and (iii) C1-open set of non-hyperbolic vector fields (that admit a Lorenz attractor). We also provide a criterion for the triviality of the C0-centralizer of continuous flows.

Rigidity of critical circle maps

We prove that any two C4 critical circle maps with the same irrational rotation number and the same odd criticality are conjugate to each other by a C1 circle diffeomorphism. The conjugacy is C1+α for a full Lebesgue measure set of rotation numbers.

Hyperbolic polygonal billiards with finitely many ergodic SRB measures

We study polygonal billiards with reflection laws contracting the angle of reflection towards the normal. It is shown that if a polygon does not have parallel sides facing each other, then the corresponding billiard map has finitely many ergodic Sinai–Ruelle–Bowen measures whose basins cover a set of full Lebesgue measure.

Generic properties of the lower spectral radius for some low-rank pairs of matrices

The lower spectral radius of a set of d×d matrices is defined to be the minimum possible exponential growth rate of long products of matrices drawn from that set. When considered as a function of a finite set of matrices of fixed cardinality it is known that the lower spectral radius can vary discontinuously as a function of the matrix entries. In a previous article the author and J. Bochi conjectured that when considered as a function on the set of all pairs of 2×2 real matrices, the lower spectral radius is discontinuous on a set of positive (eight-dimensional) Lebesgue measure, and related this result to an earlier conjecture of Bochi and Fayad. In this article we investigate the continuity of the lower spectral radius in a simplified context in which one of the two matrices is assumed to be of rank one. We show in particular that the set of discontinuities of the lower spectral radius on the set of pairs of 2×2 real matrices has positive seven-dimensional Lebesgue measure, and that among the pairs of matrices studied, the finiteness property for the lower spectral radius is true on a set of full Lebesgue measure but false on a residual set.

On the Complete Integrability of the Raychaudhuri Differential System in $$\mathbb {R}^4$$ R 4 and of a CRNT Model in $$\mathbb {R}^5$$ R 5

We study the Darboux integrability of two differential systems with parameters: the Raychaudhuri equation (a relativistic model in \(\mathbb {R}^4\)) and a chemical reaction model in \(\mathbb {R}^5\). We prove that the first one is completely integrable and that the first integrals are of Darboux type. This is the first four-dimensional realistic non-trivial model which is completely integrable with first integrals of Darboux type and for which for a full Lebesgue measure set of the values of the parameters the three linearly independent first integrals are rational. For the second one, we find all its Darboux polynomials and exponential factors and we prove that it is not Darboux integrable.

On the multifractal local behavior of parabolic stochastic PDEs

Consider the stochastic heat equation $\dot{u} =\frac 12 u''+\sigma (u)\xi $ on $(0\,,\infty )\times \mathbb{R} $ subject to $u(0)\equiv 1$, where $\sigma :\mathbb{R} \to \mathbb{R} $ is a Lipschitz (local) function that does not vanish at $1$, and $\xi $ denotes space-time white noise. It is well known that $u$ has continuous sample functions [22]; as a result, $\lim _{t\downarrow 0}u(t\,,x)= 1$ almost surely for every $x\in \mathbb{R} $. The corresponding fluctuations are also known [14, 16, 20]: For every fixed $x\in \mathbb{R} $, $t\mapsto u(t\,,x)$ looks locally like a fixed multiple of fractional Brownian motion (fBm) with index $1/4$. In particular, an application of Fubini’s theorem implies that, on an $x$-set of full Lebesgue measure, the short-time behavior of the peaks of the random function $t\mapsto u(t\,,x)$ are governed by the law of the iterated logarithm for fBm, up to possibly a suitable normalization constant. By contrast, the main result of this paper claims that, on an $x$-set of full Hausdorff dimension, the short-time peaks of $t\mapsto u(t\,,x)$ follow a non-iterated logarithm law, and that those peaks contain a rich multifractal structure a.s. Large-time variations of these results were predicted in the physics literature a number of years ago and proved very recently in [10, 11]. To the best of our knowledge, the short-time results of the present paper are observed here for the first time.

On dynamical systems perturbed by a null-recurrent motion: The general case

We consider a perturbed ordinary differential equation where the perturbation is only significant when a one-dimensional null recurrent diffusion is close to zero. We investigate the first order correction to the unperturbed system and prove a central limit theorem type result, i.e., that the normalized deviation process converges in law in the space of continuous functions to a limit process which we identify. We show that this limit process has a component which only moves when the limit of the null-recurrent fast motion equals zero. The set of these times forms a zero-measure Cantor set and therefore the limiting process cannot be described by a standard SDE. We characterize this process by its infinitesimal generator (with appropriate boundary conditions) and we also characterize the process as the weak solution of an SDE that depends on the local time of the fast motion process. We also investigate the long time behavior of such a system when the unperturbed motion is trivial. In this case, we show that the long-time limit is constant on a set of full Lebesgue measure with probability 1, but it has nontrivial drift and diffusion components that move only when the fast motion equals zero.

On the PropertyN-1

We construct a continuous functionf:[0,1]→Rsuch thatfpossessesN-1-property, butfdoes not have approximate derivative on a set of full Lebesgue measure. This shows that Banach’s Theorem concerning differentiability of continuous functions with Lusin’s property(N)does not hold forN-1-property. Some relevant properties are presented.

Existence of sectional-hyperbolic attractors for three-dimensional flows

We prove that every C1 generic three-dimensional flow has either infinitely many sinks or finitely many sectional-hyperbolic attractors whose basins form a full Lebesgue measure set. In particular, all such flows exhibit sectional-hyperbolic attractors.

Multipoint nonlocal problem for factorized equation with dependent coefficients in conditions

The conditions of correct solvability of multipoint nonlocal problem for factorized PDE with coefficients in conditions, which depends on one real parameter, is established. It is shown that these conditions on the set of fullLebesgue measure of the interval parameters are fulfilled.

Vector--valued parabolic $\omega$--minimizers

We are investigating regularity issues for parabolic almost minimizers of integral functionals with polynomial growth structure. In particular, we are interested in functions with values in the target space $\mathbb R^N$ for a dimension $N > 1$ and our aim is to prove Hölder continuity for the spatial gradient of an almost minimizer on an open set of full Lebesgue measure. As classical counter examples show, the regularity results which we achieve are optimal if we are dealing with general almost minimizers and dimensions of the target space strictly greater than one. However, the results are also valid in the case $N=1$, even if they are not optimal in that case. Our contribution can be seen as a natural generalization of the elliptic regularity theory for the gradient of vector--valued almost minimizers, which was developed in the 1980s to the framework of time-dependent, parabolic, problems. The parabolic nature of the problem, in combination with the fact that almost minimizers, in general, do not satisfy any associated parabolic partial differential equation, causes a variety of difficulties in regularity theory, which have to be overcome. The first part of the paper covers integral functionals, which involve merely the spatial gradient of the almost minimizing function. In this case, we obtain a partial Hölder regularity result for the spatial gradient with a Hölder exponent, which can be determined in a quantified way in dependence of the data of the problem. In the second part, we develop the theory for integrands, which involve not only the spatial gradient, but also the almost minimizer itself. In this more general case, the proof becomes substantially more involved, since a priori higher integrability of the spatial gradient has to be used. As a consequence, the explicit control on the Hölder continuity exponent gets lost. This is also in accordance with the elliptic theory where the same phenomenon occurs.

On Araujo’s Theorem for Flows

We prove that every C1 generic three-dimensional flow without singularities has either infinitely many sinks or finitely many hyperbolic attractors whose basins form a full Lebesgue measure set.

Distributionally scrambled invariant sets in a compact metric space

The paper solves a question posed by Oprocha on the existence of invariant distributionally chaotic scrambled sets. We show, among other things, that a continuous map f acting on compact metric space (X,d) with a weak specification property, fixed point, and infinitely many mutually distinct periods has a dense Mycielski (i.e., c dense set of type Fσ) invariant distributionally scrambled set. As a consequence, we describe a class of maps with a distributionally scrambled invariant set of full Lebesgue measure in the case when X is a k-dimensional cube.

Free path of billiards with flat points

In this paper we study a special family of Lorentz gas with infinite horizon. The periodic scatterers have $C^3$ smooth boundary with positive curvature except on finitely many flat points. In addition there exists a trajectory with infinite free path and tangentially touching the scatterers only at some flat points. The singularity set of the system is analyzed in detail. And we prove that the free path is piecewise Hölder continuous with uniform Hölder constant. In addition these systems are shown to be non-uniformly hyperbolic;local stable and unstable manifolds exist on a set of full Lebesgue measure; and the stable and unstable holonomy maps are absolutely continuous.

Level sets of the Takagi function: local level sets

The Takagi function τ: [0,1] → [0, 1] is a continuous non-differentiable function constructed by Takagi in 1903. The level sets L(y) = {x : τ(x) = y} of the Takagi function τ(x) are studied by introducing a notion of local level set into which level sets are partitioned. Local level sets are simple to analyze, reducing questions to understanding the relation of level sets to local level sets, which is more complicated. It is known that for a “generic” full Lebesgue measure set of ordinates y, the level sets are finite sets. In contrast, here it is shown for a “generic” full Lebesgue measure set of abscissas x, the level set L(τ(x)) is uncountable. An interesting singular monotone function is constructed associated to local level sets, and is used to show the expected number of local level sets at a random level y is exactly \({\frac{3}{2}}\).

Partial regularity for parabolic systems with non-standard growth

We study non-linear parabolic systems with non-standard p(z)-growth conditions and establish that the gradient of weak solutions is locally Holder continuous with Holder exponent $${\beta \in (0,1)}$$ with respect to the parabolic metric on an open set of full Lebesgue measure, provided the exponent function p(z) itself is Holder continuous with exponent β with respect to the parabolic metric.

Level sets of the Takagi function: generic level sets

The Takagi function {\tau} : [0, 1] \rightarrow [0, 1] is a continuous non-differentiable function constructed by Takagi in 1903. This paper studies the level sets L(y) = {x : {\tau}(x) = y} of the Takagi function {\tau}(x). It shows that for a full Lebesgue measure set of ordinates y, these level sets are finite sets, but whose expected number of points is infinite. Complementing this, it shows that the set of ordinates y whose level set has positive Hausdorff dimension is itself a set of full Hausdorff dimension 1 (but Lebesgue measure zero). Finally it shows that the level sets have a nontrivial Hausdorff dimension spectrum. The results are obtained using a notion of "local level set" introduced in a previous paper, along with a singular measure parameterizing such sets.

Typical points for one-parameter families of piecewise expanding maps of the interval

For one-parameter families of piecewise expanding maps of the intervalwe establish sufficient conditions such that a given point in the interval is typicalfor the absolutely continuous invariant measure for a full Lebesgue measure set of parameters.In particular, we consider $C^{1,1}(L)$-versions of $\beta$-transformations,piecewise expanding unimodal maps, and Markov structure preserving one-parameter families.For families of piecewise expanding unimodal maps we show that the turning point is almost surely typicalwhenever the family is transversal.

The solar Julia sets of basic quadratic Cremer polynomials

AbstractIn general, little is known about the exact topological structure of Julia sets containing a Cremer point. In this paper we show that there exist quadratic Cremer Julia sets of positive area such that for a full Lebesgue measure set of angles the impressions are degenerate, the Julia set is connected im kleinen at the landing points of these rays, and these points are contained in no other impression.