Abstract
AbstractA necessary and sufficient condition for the iterated function system $$\{f(\cdot ,\omega )\,|\,\omega \in \Omega \}$$
{
f
(
·
,
ω
)
|
ω
∈
Ω
}
with probability P to have exactly one invariant measure $$\mu _*$$
μ
∗
with $$\mu _*((0,1))=1$$
μ
∗
(
(
0
,
1
)
)
=
1
is given. The main novelty lies in the fact that we only require the transformations $$f(\cdot ,\omega )$$
f
(
·
,
ω
)
to be increasing homeomorphims, without any smoothness condition, neither we impose conditions on the cardinality of $$\Omega $$
Ω
. In particular, positive Lyapunov exponents conditions are replaced with the existence of solutions to some functional inequalities. The stability and strong law of large numbers of the considered system are also proven.
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Discrete Mathematics and Combinatorics
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