Inverse limits of certain interval mappings as attractors in two dimensions
Inverse problems of symbolic dynamics
This paper reviews some results regarding symbolic dynamics, correspondence between languages of dynamical systems and combinatorics. Sturmian sequences provide a pattern for investigation of one-dimensional systems, in particular interval exchange transformation. Rauzy graphs language can express many important combinatorial and some dynamical properties. In this case combinatorial properties are considered as being generated by a substitutional system, and dynamical properties are considered...
Kneading theory for a family of circle maps with one discontinuity.
Logarithmic measures, subdimension and Lyapunov exponents of Cantori
Maps of the interval Ljapunov stable on the set of nonwandering points.
Metric diophantine approximation in Julia sets of expanding rational maps
Minor cycles for interval maps
For continuous maps of an interval into itself we consider cycles (periodic orbits) that are non-reducible in the sense that there is no non-trivial partition into blocks of consecutive points permuted by the map. Among them we identify the miror ones. They are those whose existence does not imply existence of other non-reducible cycles of the same period. Moreover, we find minor patterns of a given period with minimal entropy.
Monotonicity properties of some skew tent maps
Nombre de rotation, structures géométriques sur un cercle et groupe de Bott-Virasoro
Une classification complète des stabilisateurs coadjoints du groupe de Bott-Virasoro est obtenue par une méthode essentiellement géométrique. L’outil de base est le nombre de rotation d’un difféomorphisme du cercle. En particulier, nous mettons en évidence la présence de groupes d’isotropie non-connexes et montrons que la transformation de Miura des opérateurs de Hill peut s’interpréter comme une application moment sur l’espace des structures affines du cercle.
Noninvertible minimal maps
For a discrete dynamical system given by a compact Hausdorff space X and a continuous selfmap f of X the connection between minimality, invertibility and openness of f is investigated. It is shown that any minimal map is feebly open, i.e., sends open sets to sets with nonempty interiors (and if it is open then it is a homeomorphism). Further, it is shown that if f is minimal and A ⊆ X then both f(A) and share with A those topological properties which describe how large a set is. Using these results...
On a one-dimensional analogue of the Smale horseshoe
We construct a transformation T:[0,1] → [0,1] having the following properties: 1) (T,|·|) is completely mixing, where |·| is Lebesgue measure, 2) for every f∈ L¹ with ∫fdx = 1 and φ ∈ C[0,1] we have , where μ is the cylinder measure on the standard Cantor set, 3) if φ ∈ C[0,1] then for Lebesgue-a.e. x.
On fixed points of holomorphic type
We study a linearization of a real-analytic plane map in the neighborhood of its fixed point of holomorphic type. We prove a generalization of the classical Koenig theorem. To do that, we use the well known results concerning the local dynamics of holomorphic mappings in ℂ².
On generic chaos of shifts in function spaces
On tame embeddings of solenoids into 3-space
Solenoids are inverse limits of the circle, and the classical knot theory is the theory of tame embeddings of the circle into 3-space. We make a general study, including certain classification results, of tame embeddings of solenoids into 3-space, seen as the "inverse limits" of tame embeddings of the circle. Some applications in topology and in dynamics are discussed. In particular, there are tamely embedded solenoids Σ ⊂ ℝ³ which are strictly achiral. Since solenoids are non-planar,...
On the arithmetic sum of regular Cantor sets
On the forcing relation for surface homeomorphisms
On the ∗-product in kneading theory
We discuss a generalization of the *-product in kneading theory to maps with an arbitrary finite number of turning points. This is based on an investigation of the factorization of permutations into products of permutations with some special properties relevant for dynamics on the unit interval.
On the range of fractional parts {ξ(p/q)ⁿ}
On the structure of minimal attraction centers of recurrent trajectories of continuous maps of the interval.
On the topology of solenoidal attractors of the cylinder