Nonhyperbolic persistent attractors near the Morse–Smale boundary

Let $$\lambda _{1},\ldots ,\lambda _{n}$$ be real numbers in $$(0,1)$$ and $$p_{1},\ldots ,p_{n}$$ be points in $$\mathbb {R}^{d}$$ . Consider the collection of maps $$f_{j}:\mathbb {R}^{d}\rightarrow \mathbb {R}^{d} $$ given by $$\begin{aligned} f_{j}(x)=\lambda _{j} x +\left( 1-\lambda _{j}\right) p_{j}. \end{aligned}$$ It is a well known result that there exists a unique nonempty compact set $$\Lambda \subset \mathbb {R}^{d}$$ satisfying $$\Lambda =\cup _{j=1}^{n} f_{j}(\Lambda ).$$ Each $$x\in \Lambda $$ has at least one coding, that is a sequence $$(\epsilon _{i})_{i=1}^{\infty }\in \{1,\ldots ,n\}^{\mathbb {N}}$$ that satisfies $$\lim _{N\rightarrow \infty }f_{\epsilon _{1}}\ldots f_{\epsilon _{N}} (0)=x.$$ We study the size and complexity of the set of codings of a generic $$x\in \Lambda $$ when $$\Lambda $$ has positive Lebesgue measure. In particular, we show that under certain natural conditions almost every $$x\in \Lambda $$ has a continuum of codings. We also show that almost every $$x\in \Lambda $$ has a universal coding. Our work makes no assumptions on the existence of holes in $$\Lambda $$ and improves upon existing results when it is assumed $$\Lambda $$ contains no holes.

It is natural to expect that the arithmetic sum of two Cantor sets should have positive Lebesgue measure if the sum of their dimensions exceeds 1, but there are many known counterexamples, e.g. when both sets are the middle-$\alpha$ Cantor set and $\alpha \in ({1 \over 3}, \frac 12)$. We show that for any compact set $K$ and for a.e. $\alpha \in (0,1)$, the arithmetic sum of $K$ and the middle-$\alpha$ Cantor set does indeed have positive Lebesgue measure when the sum of their Hausdorff dimensions exceeds 1. In this case we also determine the essential supremum, as the translation parameter $t$ varies, of the dimension of the intersection of $K+t$ with the middle-$\alpha$ Cantor set. We also establish a new property of the infinite Bernoulli convolutions $\nu _\lambda ^p$ (the distributions of random series $\sum _{n=0}^\infty \pm \lambda ^n ,$ where the signs are chosen independently with probabilities $(p,1-p)$). Let $1 \leq q_1<q_2 \leq 2$. For $p \neq \frac 12$ near $\frac 12$ and for a.e. $\lambda$ in some nonempty interval, $\nu _\lambda ^p$ is absolutely continuous and its density is in $L^{q_1}$ but not in $L^{q_2}$. We also answer a question of Kahane concerning the Fourier transform of $\nu _\lambda$.

T. J. Willmore has shown that if a differentiable manifold's rank (the maximum number of everywhere linearly independent commuting vector fields definable on it) equals the manifold's dimension, then the manifold is a torus of the appropriate dimension [1]. This theorem is proved more simply and without any differentiability hypothesis in the present note.

Differentiability of continuous functions in terms of Haar-smallness

Hyperbolic relative equilibria for the negative curved n–body problem

We show that there exists a transcendental entire function whose Julia set has positive finite Lebesgue measure.

We know from Netto’s theorem (Theorem 6.4) that plane curves that do not cross (or touch) themselves cannot be space-filling. We will show in this chapter that it is still possible for such curves to have a positive two-dimensional Lebesgue measure.

For λ ∈ [0.6439, 0.6441]∪[0.6458, 0.6466]∪[0.6470, 0.6472], the “fat” Sierpiński triangle—the unique compact invariant set (“attractor”) Λλ of the iterated function system on , where p0 = (0, 0), p1 = (1, 0), p2 = (1/2, 1)—has interior points and hence positive Lebesgue measure and Hausdorff dimension 2. For these λ, standard techniques for determining the Hausdorff dimension (or Lebesgue measure) of Λλ do not apply, and its Hausdorff dimension has not been known for any specific such λ. The novelty of our approach is that instead of extending techniques developed for small λ, we significantly extend geometric methods developed by Broomhead, Montaldi, and Sidorov for larger values of λ.

Let us denote the nth difference between consecutive primes by d_n. The Prime Number Theorem clearly implies that d_n is logn on average. Paul Erd\H{o}s conjectured about 60 years ago that the sequence d_n/logn is everywhere dense on the nonnegative part of the real line. He and independently G. Ricci proved in 1954-55 that the set J of limit points of the sequence {d_n/logn} has positive Lebesgue measure. The first and until now only concrete known element of J was proved to be the number zero in the work of Goldston, Yildirim and the present author. The author of the present note showed in 2013 (arXiv: 1305.6289) that there is a fixed interval containing 0 such that all elements of it are limit points. In 2014 it was shown by W.D. Banks, T. Freiberg and J. Maynard (arXiv: 1404.5094) that one can combine the Erd\H{o}s-Rankin method (producing large prime gaps) and the Maynard-Tao method (producing bounded prime gaps) to obtain the lower bound (1+o(1))T/8 for the Lebesgue measure of the subset of limit points not exceeding T. In the present note we improve this lower bound to (1+o(1))T/4, using a refinement of the argument of Banks, Freiberg and Maynard.

A continuous function, chaotic in the sense of Li and Yorke, is constructed which possesses a scrambled set of positive Lebesgue measure.

Introduction. This paper is an announcement of a study on the theory of the topological variation of the phase space of one-parameter families of vector fields (ordinary differential equations, flows). This theory, sometimes called bifurcation theory, has been developed since H. Poincare from several points of view; see, for example, [l]> [z]i [3]» W> [5]. Here, we will be mainly interested in a collection of one-parameter families of vector fields which has the following properties: (a) it is large with respect to all the families, and (b) its elements exhibit a topological variation which is amenable to simple description. Collections with properties (a) and (b) are currently called generic and were introduced in the qualitative global analysis of differential equations by M. Peixoto [o], S. Smale [8], and I. Kupka [ l l ] . See S. Smale [9] for a thorough monography on this field. The geometry of the set 2 of structurally stable vector fields and the study of generic one-parameter families of vector fields are closely related. A vector field is structurally stable if its phase space does not change topologically under small perturbations; a oneparameter family of vector fields exhibits: the simpler a phase-space topological variation, the larger the intersection it has with 2 , or equivalently, the smaller the intersection it has with its complement —the set of nonstructurally stable vector fields. The importance of the set of nonstructurally stable vector fields for the study of the topological variation of the phase space of vector fields was noticed by A. Andronov and E. Leontovich. In [12] they defined the concept of first-order structural stability as a possible guide to pursue such study. In compact two-dimensional manifolds, the only case considered here, the set of first-order structurally stable vector fields is an imbedded Banach manifold of class C and codimension one of the Banach manifold of vector fields; see [13]. This fact, although (as follows from [14]) not sufficient to describe completely the generic

Let [Formula: see text] be a [Formula: see text] expanding map of the circle and let [Formula: see text] be a [Formula: see text] function. Consider the twisted cohomological equation [Formula: see text] which has a unique bounded solution [Formula: see text]. We show that [Formula: see text] is either [Formula: see text] or continuous but nowhere differentiable. If [Formula: see text] is nowhere differentiable then the Newton quotients of [Formula: see text], after an appropriated normalization, converges in distribution (with respect to the unique absolutely continuous invariant probability of [Formula: see text]) to the normal distribution. In particular, [Formula: see text] is not a Lipschitz continuous function on any subset with positive Lebesgue measure.

International audience

Bounded point evaluations and capacity

In this paper, we study the existence and concentration of positive solution of a class of coupled Schrödinger equations. We admit that the potentials may not be non-negative and suppose that the intersection of the sets has positive Lebesgue measure. By studying the modified functional of the associated functional carefully, we establish the existence of positive least energy solutions for the coupled Schrödinger system. Moreover, we prove the concentration phenomenon of the positive solution when the parameter goes to infinity.