17 research outputs found
The recurrence function of a random Sturmian word
This paper describes the probabilistic behaviour of a random Sturmian word.
It performs the probabilistic analysis of the recurrence function which can be
viewed as a waiting time to discover all the factors of length of the
Sturmian word. This parameter is central to combinatorics of words. Having
fixed a possible length for the factors, we let to be drawn
uniformly from the unit interval , thus defining a random Sturmian word
of slope . Thus the waiting time for these factors becomes a random
variable, for which we study the limit distribution and the limit density.Comment: Submitted to ANALCO 201
Dynamics of Modular Matings
In the paper 'Mating quadratic maps with the modular group II' the current
authors proved that each member of the family of holomorphic
correspondences :
introduced by the first author and C.
Penrose in 'Mating quadratic maps with the modular group', is a mating between
the modular group and a member of the parabolic family of quadratic rational
maps whenever the limit set of is connected.
Here we provide a dynamical description for the correspondences
which parallels the Douady and Hubbard description for
quadratic polynomials. We define a B\"ottcher map and a Green's function for
, and we show how in this setting periodic geodesics play the
role played by external rays for quadratic polynomials. Finally, we prove a
Yoccoz inequality which implies that for the parameter to be in the
connectedness locus of the family , the value of
the log-multiplier of an alpha fixed point which has combinatorial rotation
number lies in a strip whose width goes to zero at rate proportional to
Dynamics of continued fractions and kneading sequences of unimodal maps
In this paper we construct a correspondence between the parameter spaces of
two families of one-dimensional dynamical systems, the alpha-continued fraction
transformations T_alpha and unimodal maps. This correspondence identifies
bifurcation parameters in the two families, and allows one to transfer
topological and metric properties from one setting to the other. As an
application, we recover results about the real slice of the Mandelbrot set, and
the set of univoque numbers.Comment: 21 pages, 3 figures. New section added with additional results and
applications. Figures and references added. Introduction rearrange
Resolution of an integral equation with the Thue-Morse sequence
It is a classical fact that the exponential function is solution of the
integral equation . If we slightly modify this
equation to with , it
seems that no classical techniques apply to yields solutions. In this article,
we consider the parameter . We will show the existence of a
solution wich takes the values of the Thue-Morse sequence on the odd integers.Comment: 9 page
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Notions of complexity in substitution dynamical systems
There has been a lot of work done in recent decades in the field of symbolic dynamics.
Much attention has been paid to the so-called "complexity" function, which gives a sense
of the rate at which the number of words in the system grow. In this paper, we explore this
and several notions of complexity of specific symbolic dynamical systems. In particular,
we compute positive entropy and state some k-balancedness properties of a few specific
(random) substitutions. We also view certain sequences as subsets of Z², stating several
properties and computing bounds on entropy in a specific example
S-adic characterization of minimal ternary dendric shifts
Dendric shifts are defined by combinatorial restrictions of the extensions of the words
in their languages. This family generalizes well-known families of shifts such as Sturmian
shifts, Arnoux-Rauzy shifts and codings of interval exchange transformations. It is known
that any minimal dendric shift has a primitive S-adic representation where the morphisms
in S are positive tame automorphisms of the free group generated by the alphabet. In
this paper we investigate those S-adic representations, heading towards an S-adic characterization
of this family. We obtain such a characterization in the ternary case, involving
a directed graph with 2 vertices
The Nevai Condition
We study Nevai's condition that for orthogonal polynomials on the real line,
where is the CD
kernel. We prove that it holds for the Nevai class of a finite gap set
uniformly on the spectrum and we provide an example of a regular measure on
where it fails on an interval