A cooperative system which does not satisfy the limit set dichotomy

Mittag-Leffler Dichotomy and Roughness of Conformable Fractional Differential Equations

The solutions of traditional fractional differential equations neither satisfy group property nor generate dynamical systems, so the study on hyperbolicity is in blank. Relying on the new proposed conformable fractional derivative, we investigate dichotomy of conformable fractional equations, including structure of solutions of linear systems, Mittag-Leffler dichotomy and stability, roughness and nonuniform dichotomy.

Looking for a Billiard Table which is not a Lattice Polygon but Satisfies Veech's Dichotomy

Over the course of studying billiard dynamics, several questions were raised. One of the questions was, which surfaces satisfy the following property (which is called Veech's dichotomy): Any direction is either completely periodic or uniquely ergodic. In an important paper Veech gave a sufficient condition for this dichotomy. He showed that if the stabilizer of a translation surface is a lattice in $SL_2(\R)$, then the surface satisfies Veech's dichotomy. Later, Smillie and Weiss proved that this condition is not necessary. They constructed a translation surface which satisfies Veech's dichoto…

A dichotomy of self-conformal subsets of the real line with overlaps

We show that self-conformal subsets of $\mathbb{R}$ that do not satisfy the weak separation condition have full Assouad dimension. Combining this with a recent results by Käenmäki and Rossi we conclude that an interesting dichotomy applies to self-conformal and not just self-similar sets: if $F\subset\mathbb{R}$ is self-conformal with Hausdorff dimension strictly less than $1$, either the Hausdorff dimension and Assouad dimension agree or the Assouad dimension is $1$. We conclude that the weak separation property is in this case equivalent to Assouad and Hausdorff dimension coinciding. (This m…

Non-Atkinson perturbations of nonautonomous linear hamiltonian systems: exponential dichotomy and nonoscillation

We analyze the presence of exponential dichotomy (ED) and of global existence of Weyl functions $M^\pm$ for one-parametric families of finite-dimensional nonautonomous linear Hamiltonian systems defined along the orbits of a compact metric space, which are perturbed from an initial one in a direction which does not satisfy the classical Atkinson condition: either they do not have ED for any value of the parameter; or they have it for at least all the nonreal values, in which case the Weyl functions exist and are Herglotz. When the parameter varies in the real line, and if the unperturbed famil…

图定向吸引子自相似性的二元性

A dichotomy on the self-similarity of graph-directed attractors

本文研究了图定向吸引子自相似性的二元性，证明了在特定条件下，某些图定向迭代函数系统的吸引子无法由标准迭代函数系统再现。