98 research outputs found
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
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
Asymptotically periodic piecewise contractions of the interval
We consider the iterates of a generic injective piecewise contraction of the
interval defined by a finite family of contractions. Let , , be -diffeomorphisms with whose images are
pairwise disjoint. Let and let be a
partition of the interval into subintervals having interior
, where and . Let be the
map given by if , for . Among other
results we prove that for Lebesgue almost every , the
piecewise contraction is asymptotically periodic.Comment: 8 page
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