Related papers: On Ilyashenko's Statistical Attractors

Metabelian algebras are introduced and it is shown that an algebra $A$ is metabelian if and only if $A$ is a nilpotent algebra having the index of nilpotency at most $3$, i.e. $x y z t = 0$, for all $x$, $y$, $z$, $t \in A$. We prove that…

We consider the robust family of Geometric Lorenz attractors. These attractors are chaotic in the sense that they are transitive and have sensitive dependence on the initial conditions. Moreover, they support SRB measures whose ergodic…

We prove that in a Euclidean space of dimension at least two, there exists a compact set of Lebesgue measure zero such that any real-valued Lipschitz function defined on the space is differentiable at some point in the set. Such a set is…

Suppose that $\mathcal{C}$ is the space of all middle Cantor sets. We characterize all triples $(\alpha,~\beta,~\lambda)\in \mathcal{C}\times\mathcal{C}\times \mathbb{R}^*$ that satisfy $C_\alpha- \lambda C_\beta=[-\lambda,~1]. $ Also all…

A version of the Riesz-Sobolev convolution inequality is formulated and proved for arbitrary compact connected Abelian groups. Maximizers are characterized and a quantitative stability theorem is proved, under natural hypotheses. A…

We give a new definition for a Lyapunov exponent (called new Lyapunov exponent) associated to a continuous map. Our first result states that these new exponents coincide with the usual Lyapunov exponents if the map is differentiable. Then,…

We describe a general program for studying the dynamics of surjective endomorphisms of algebraic varieties that are amenable to techniques from the minimal model program. We obtain density results on the pre-periodic points of surjective…

Weakly-irreducible not irreducible subalgebras of $\so(1,n+1)$ were classified by L. Berard Bergery and A. Ikemakhen. In the present paper a geometrical proof of this result is given. Transitively acting isometry groups of Lobachevskian…

We discuss the use of Kaniadakis' $\kappa$-exponential in the construction of a statistical manifold modelled on Lebesgue spaces of real random variables. Some algebraic features of the deformed exponential models are considered. A chart is…

Let G be a possibly disconnected reductive group over a finite field with Frobenius map F. The main result of this paper is that the characteristic functions af "admissible complexes" A on G such that F^*A is isomorphic to A form a basis of…

For any integer $r$ with $1\leq r<\infty$, we present a one-parameter family $F_\sigma$ $(0<\sigma<1)$ of 2-dimensional piecewise $\mathcal C^r$ expanding maps such that each $F_\sigma$ has an observable (i.e. Lebesgue positive) Lyapunov…

There are few examples of non-autonomous vector fields exhibiting complex dynamics that may be proven analytically. We analyse a family of periodic perturbations of a weakly attracting robust heteroclinic network defined on the two-sphere.…

We establish the well-posedness of a strongly damped semilinear wave equation equipped with nonlinear hyperbolic dynamic boundary conditions. Results are carried out with the presence of a parameter distinguishing whether the underlying…

We study the spectral properties of positive absolutely minimum attaining operators defined on infinite dimensional complex Hilbert spaces and using that derive a characterization theorem for such type of operators. We construct several…

The evoluted set is the set of configurations reached from an initial set via a fixed flow for all times in a fixed interval. We find conditions on the initial set and on the flow ensuring that the evoluted set has negligible boundary (i.e.…

The present paper is devoted to a study of orientation-preserving homeomorphisms on three-dimensional manifolds with a non-wandering set consisting of a finite number of surface attractors and repellers. The main results of the paper relate…

We show that linear analytic cocycles where all Lyapunov exponents are negative infinite are nilpotent. For such one-frequency cocycles we show that they can be analytically conjugated to an upper triangular cocycle or a Jordan normal form.…

We study algorithmically random closed subsets of $2^\omega$, algorithmically random continuous functions from $2^\omega$ to $2^\omega$, and algorithmically random Borel probability measures on $2^\omega$, especially the interplay between…

We investigate the topological and metric properties of attractors of an iterated function system (IFS) whose functions may not be contractive. We focus, in particular, on invertible IFSs of finitely many maps on a compact metric space. We…

Let $E$ be a linear space and suppose that $A$ is the global attractor of either (i) a homeomorphism $F:E\rightarrow E$ or (ii) a semigroup $S(\cdot)$ on $E$ that is injective on $A$. In both cases $A$ has trivial shape, and the dynamics on…