We show that, for the family of functions $F_{\lambda }\left(z\right)=zⁿ+\lambda /zⁿ$ where n ≥ 3 and λ ∈ ℂ, there is a unique McMullen domain in parameter space. A McMullen domain is a region where the Julia set of $F_{\lambda }$ is homeomorphic to a Cantor set of circles. We also prove that this McMullen domain is a simply connected region in the plane that is bounded by a simple closed curve.

In this paper we describe a $2$-dimensional generalization of the Euclidean algorithm which stems from the dynamics of $3$-interval exchange transformations. We investigate various diophantine properties of the algorithm including the quality of simultaneous approximations. We show it verifies the following Lagrange type theorem: the algorithm is eventually periodic if and only if the parameters lie in the same quadratic extension of $\mathbb{Q}.$

Soit $M$ une variété différentiable de dimension paire munie d’une 2-forme différentielle fermée générique $\Omega$. L’apparition éventuelle d’un lieu de dégénérescence $\Sigma \left(\Omega \right)$ du rang de $\Omega$ est l’obstacle à ce que $(M,\Omega )$ soit une structure symplectique. Nous étudions les propriétés géométriques de $\Sigma \left(\Omega \right)$ et nous caractérisons l’algèbre des hamiltoniennes admissibles de $(M,\Omega )$ i.e. les fonctions différentiables $h$ qui possèdent un champ hamiltonien $X_h$ sur $M$.

We give analogs of the complexity $p\left(n\right)$ and of Sturmian words which are called respectively the $*$-complexity $p_{*}\left(n\right)$ and $*$-Sturmian words. We show that the class of $*$-Sturmian words coincides with the class of words satisfying $p_{*}\left(n\right)\le n+1$, and we determine the structure of $*$-Sturmian words. For a class of words satisfying $p_{*}\left(n\right)=n+1$, we give a general formula and an upper bound for $p\left(n\right)$. Using this general formula, we give explicit formulae for $p\left(n\right)$ for some words belonging to this class. In general, $p\left(n\right)$ can take large values, namely,...

We investigate properties of the zero of the subadditive pressure which is a most important tool to estimate the Hausdorff dimension of the attractor of a non-conformal iterated function system (IFS). Our result is a generalization of the main results of Miao and Falconer [Fractals 15 (2007)] and Manning and Simon [Nonlinearity 20 (2007)].

We discuss the inverse limit spaces of unimodal interval maps as topological spaces. Based on the combinatorial properties of the unimodal maps, properties of the subcontinua of the inverse limit spaces are studied. Among other results, we give combinatorial conditions for an inverse limit space to have only arc+ray subcontinua as proper (non-trivial) subcontinua. Also, maps are constructed whose inverse limit spaces have the inverse limit spaces of a prescribed set of periodic unimodal maps as...

In this paper we study the existence of subharmonic solutions of the hamiltonian system$$J\dot{x}+u^{*}\nabla G(t,u\left(x\right))=e\left(t\right)$$where $u$ is a linear map, $G$ is a $C^1$-function and $e$ is a continuous function.

In this paper we study the existence of subharmonic solutions of the Hamiltonian system $$J\dot{x}+u^{*}\nabla G(t,u\left(x\right))=e\left(t\right)$$ where u is a linear map, G is a C1-function and e is a continuous function.

We give a few examples of substitutions on infinite alphabets, and the beginning of a general theory of the associated dynamical systems. In particular, the “drunken man” substitution can be associated to an ergodic infinite measure preserving system, of Krengel entropy zero, while substitutions of constant length with a positive recurrent infinite matrix correspond to ergodic finite measure preserving systems.