scholarly journals Strongly Recurrent Transformation Groups

2021 ◽  
Author(s):  
Massoud Amini ◽  
Jumah Swid
Keyword(s):  
Positive Measure ◽  
Invariant Measures ◽  
Transformation Groups ◽  
Ordered Groups ◽  
Atomic Measure ◽  
Measure Spaces

We define the notions of strong and strict recurrency for actions of countable ordered groups on $\sigma$-finite non atomic measure spaces with quasi-invariant measures. We show that strong recurrency is equivalent to non existence of weakly wandering sets of positive measure. We also show that for certain p.m.p ergodic actions the system is not strictly recurrent, which shows that strong and strict recurrency are not equivalent.

Strongly equivalent invariant measures

1980 ◽  
Vol 88 (1) ◽  
pp. 33-43 ◽  
Author(s):  
J. Rosenblatt
Keyword(s):  
Measure Space ◽  
Positive Measure ◽  
Invariant Measures ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Zero Measure ◽  
Infinite Measure ◽  
Two Measures ◽  
Necessary And Sufficient

AbstractTwo measures are strongly equivalent if they have the same sets of zero measure and the same sets of infinite measure. Given a group G of strongly non-singular measurable transformations of a non-atomic positive measure space (X, β, p), if G is amenable, then a necessary and sufficient condition for there to be a G-invariant positive measure on (X, β) which is strongly equivalent to p is that p(E) > 0 implies inf p(gE) > 0 and also p(E) < ∞ implies

On universal semiregular invariant measures

1988 ◽  
Vol 53 (4) ◽  
pp. 1170-1176
Author(s):  
Piotr Zakrzewski
Keyword(s):  
Positive Measure ◽  
Invariant Measures ◽  
Finite Measure

AbstractWe consider countably additive, nonnegative, extended real-valued measures which vanish on singletons. Such a measure is universal on a set X iff it is defined on all subsets of X and is semiregular iff every set of positive measure contains a subset of positive finite measure. We study the problem of existence of a universal semiregular measure on X which is invariant under a given group of bijections of X. Moreover we discuss some properties of universal, semiregular, invariant measures on groups.

Absolutely continuous invariant measures and random perturbations for certain one-dimensional maps

1992 ◽  
Vol 12 (1) ◽  
pp. 13-37 ◽  
Author(s):  
Michael Benedicks ◽  
Lai-Sang Young
Keyword(s):  
Positive Measure ◽  
Invariant Measures ◽  
Random Perturbations ◽  
Continuous Invariant Measure ◽  
Absolutely Continuous ◽  
One Dimensional ◽  
Absolutely Continuous Invariant Measure ◽  
Small Random Perturbations ◽  
One Dimensional Maps ◽  
Absolutely Continuous Invariant

AbstractWe study the quadratic family and show that for a positive measure set of parameters the map has an absolutely continuous invariant measure that is stable under small random perturbations.

ORDER AND CHAOS FOR A CLASS OF PIECEWISE LINEAR MAPS

1995 ◽  
Vol 05 (05) ◽  
pp. 1379-1394 ◽  
Author(s):  
VÍCTOR JIMÉNEZ LÓPEZ
Keyword(s):  
Positive Measure ◽  
Invariant Measures ◽  
Initial Conditions ◽  
Piecewise Linear ◽  
Absolutely Continuous ◽  
Linear Maps ◽  
Sensitive Dependence ◽  
Piecewise Linear Maps ◽  
Almost All ◽  
Absolutely Continuous Invariant

For a class of piecewise linear maps f: I → I from a compact interval I into itself, we describe the asymptotic behavior of the sequence [Formula: see text] for almost all x ∈ I. We also study in this setting the relations among sensitive dependence on initial conditions, existence of scrambled sets of positive measure and existence of absolutely continuous invariant measures.

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