179 research outputs found
Approximation to points in the plane by SL(2,Z)-orbits
Let x be a point in R^2 with irrational slope and let \Gamma denote the
lattice SL(2,Z) acting linearly on R^2. Then, the orbit \Gamma x is dense in
R^2. We give efective results on the approximation of a point y in R^2 by
points of the form \gamma x, where \gamma belongs to \Gamma, in terms of the
size of \gamma
Absorbing sets of homogeneous subtractive algorithms
We consider homogeneous multidimensional continued fraction algorithms, in
particular a family of maps which was introduced by F. Schweiger. We prove his
conjecture regarding the existence of an absorbing set for those maps. We also
establish that their renormalisations are nonergodic which disproves another
conjecture due to Schweiger. Other homogeneous algorithms are also analysed
including ones which are ergodic
Classical homogeneous multidimensional continued fraction algorithms are ergodic
Homogeneous continued fraction algorithms are multidimensional
generalizations of the classical Euclidean algorithm, the dissipative map
(x_1,x_2) \in \mathbb{R}_+^2 \longmapsto \left\{\begin{array}{ll}
(x_1 - x_2, x_2), & \mbox{if $x_1 \geq x_2$}
(x_1, x_2 - x_1), & \mbox{otherwise.} \end{array} \right. We focus on
those which act piecewise linearly on finitely many copies of positive cones
which we call Rauzy induction type algorithms.
In particular, a variation Selmer algorithm belongs to this class. We prove
that Rauzy induction type algorithms, as well as Selmer algorithms, are ergodic
with respect to Lebesgue measure
Dynamics of piecewise contractions of the interval
We study the asymptotical behaviour of iterates of piecewise contractive maps
of the interval. It is known that Poincar\'e first return maps induced by some
Cherry flows on transverse intervals are, up to topological conjugacy,
piecewise contractions. These maps also appear in discretely controlled
dynamical systems, describing the time evolution of manufacturing process
adopting some decision-making policies. An injective map is
a {\it piecewise contraction of intervals}, if there exists a partition of
the interval into intervals ,..., such that for every
, the restriction is -Lipschitz for some
. We prove that every piecewise contraction of
intervals has at most periodic orbits. Moreover, we show that every
piecewise contraction is topologically conjugate to a piecewise linear
contraction
Multi-dimensional metric approximation by primitive points
We refine metrical statements in the style of the Khintchine-Groshev Theorem
by requiring certain coprimality constraints on the coordinates of the integer
solutions
Piecewise contractions defined by iterated function systems
Let be Lipschitz contractions. Let
, and . We prove that for Lebesgue almost every
satisfying , the piecewise
contraction defined by is
asymptotically periodic. More precisely, has at least one and at most
periodic orbits and the -limit set is a periodic orbit of
for every .Comment: 16 pages, two figure
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