26 research outputs found
Various questions around finitely positively expansive dynamical systems
It is well-known that when a positively expansive dynamical system is
invertible then its underlying space is finite. C.Morales has introduced a
decade ago a natural way to generalize positive expansiveness, by introducing
other properties that he called positive -expansiveness, for all ,
positive -expansiveness being identical to positive expansiveness. Contrary
to positive expansiveness, positive -expansiveness for does not
enforce that the space is finite when the system is invertible. In the present
paper we call finitely positively expansive dynamical systems as the ones which
are positively -expansive for some integer , and prove several results on
this class of systems. In particular, the well-known result quoted above is
true if we add the constraint of shadowing property, while it is not if this
property is replaced with minimality. Furthermore, finitely positively
expansive systems cannot occur on certain topological spaces such as the
interval, when the system is assumed to be invertible finite positive
expansiveness implies zero topological entropy. Overall we show that the class
of finitely positively expansive dynamical systems is quite rich and leave
several questions open for further research
Algorithmic Complexity for the Realization of an Effective Subshift By a Sofic
Realization of d-dimensional effective subshifts as projective sub-actions of d + d\u27-dimensional sofic subshifts for d\u27 >= 1 is now well known [Hochman, 2009; Durand/Romashchenko/Shen, 2012; Aubrun/Sablik, 2013]. In this paper we are interested in qualitative aspects of this realization. We introduce a new topological conjugacy invariant for effective subshifts, the speed of convergence, in view to exhibit algorithmic properties of these subshifts in contrast to the usual framework that focuses on undecidable properties
-adic expansions related to continued fractions (Natural extension of arithmetic algorithms and S-adic system)
"Natural extension of arithmetic algorithms and S-adic system". July 20~24, 2015. edited by Shigeki Akiyama. The papers presented in this volume of RIMS Kôkyûroku Bessatsu are in final form and refereed.We consider S-adic expansions associated with continued fraction algorithms, where an S-adic expansion corresponds to an infinite composition of substitutions. Recall that a substitution is a morphism of the free monoid. We focus in particular on the substitutions associated with regular continued fractions (Sturmian substitutions), and with Arnoux-Rauzy, Brun, and Jacobi{Perron (multidimensional) continued fraction algorithms. We also discuss the spectral properties of the associated symbolic dynamical systems under a Pisot type assumption
Dynamical Directions in Numeration
International audienceWe survey definitions and properties of numeration from a dynamical point of view. That is we focuse on numeration systems, their associated compactifications, and the dynamical systems that can be naturally defined on them. The exposition is unified by the notion of fibred numeration system. A lot of examples are discussed. Various numerations on natural, integral, real or complex numbers are presented with a special attention payed to beta-numeration and its generalisations, abstract numeration systems and shift radix systems. A section of applications ends the paper