We show that on a dense open set of analytic one-frequency complex valued cocycles in arbitrary dimension Oseledets filtration is either dominated or trivial. The underlying mechanism is different from that of the Bochi-Viana Theorem for continuous cocycles, which links non-domination with discontinuity of the Lyapunov exponent. Indeed, in our setting the Lyapunov exponents are shown to depend continuously on the cocycle, even if the initial irrational frequency is allowed to vary. On the other...

MSC 2010: 26A33, 34D05, 37C25In the paper, long-time behavior of solutions of autonomous two-component incommensurate fractional dynamical systems with derivatives in the Caputo sense is investigated. It is shown that both the characteristic times of the systems and the orders of fractional derivatives play an important role for the instability conditions and system dynamics. For these systems, stationary solutions can be unstable for wider range of parameters compared to ones in the systems with...

We establish a relationship between the word complexity and the number of generalized diagonals for a polygonal billiard. We conclude that in the rational case the complexity function has cubic upper and lower bounds. In the tiling case the complexity has cubic asymptotic growth.

Let $T:x\mapsto x+g$ be an ergodic translation on the compact group $C$ and $M\subseteq C$ a continuity set, i.e. a subset with topological boundary of Haar measure 0. An infinite binary sequence $\mathbf{a}:\mathbb{Z}\mapsto \{0,1\}$ defined by $\mathbf{a}\left(k\right)=1$ if $T^k\left(0_C\right)\in M$ and $\mathbf{a}\left(k\right)=0$ otherwise, is called a Hartman sequence. This paper studies the growth rate of $P_{\mathbf{a}}\left(n\right)$, where $P_{\mathbf{a}}\left(n\right)$ denotes the number of binary words of length $n\in \mathbb{N}$ occurring in $\mathbf{a}$. The growth rate is always subexponential and this result is optimal. If $T$ is an ergodic translation $x\mapsto x+\alpha$$...$

We study the complexity of the infinite word $u_{\beta }$ associated with the Rényi expansion of $1$ in an irrational base $\beta \>1$. When $\beta$ is the golden ratio, this is the well known Fibonacci word, which is sturmian, and of complexity $ℂ\left(n\right)=n+1$. For $\beta$ such that $d_{\beta }\left(1\right)=t_1{t}_2\cdots t_m$ is finite we provide a simple description of the structure of special factors of the word $u_{\beta }$. When $t_m=1$ we show that $ℂ\left(n\right)=(m-1)n+1$. In the cases when $t_1=t_2=\cdots =t_{m-1}$ or $t_1\>max\{t_2,\cdots ,t_{m-1}\}$ we show that the first difference of the complexity function $ℂ(n+1)-ℂ\left(n\right)$ takes value in $\{m-1,m\}$ for every $n$, and consequently we determine...

We study the complexity of the infinite word uβ associated with the Rényi expansion of 1 in an irrational base β > 1. When β is the golden ratio, this is the well known Fibonacci word, which is Sturmian, and of complexity C(n) = n + 1. For β such that dβ(1) = t1t2...tm is finite we provide a simple description of the structure of special factors of the word uβ. When tm=1 we show that C(n) = (m - 1)n + 1. In the cases when t1 = t2 = ... tm-1or t1 > max{t2,...,tm-1} we show that the first difference of...

Traitant la série de Poincaré d’un groupe discret d’isométries en courbure négative comme un noyau de Green, on établit une théorie du potentiel assez comparable à la théorie classique pour affirmer un parallèle entre densités conformes à la Patterson-Sullivan et densités harmoniques, et notamment définir une frontière de Martin où les densités ergodiques forment la partie minimale, et enfin l’identifier géométriquement sous hypothèse d’hyperbolicité.