TY  - JOUR
A1  - Gerlach, Moritz Reinhardt
T1  - Convergence of dynamics and the Perron-Frobenius operator
JF  - Israel Journal of Mathematics
N2  - We complete the picture how the asymptotic behavior of a dynamical system is reflected by properties of the associated Perron-Frobenius operator. Our main result states that strong convergence of the powers of the Perron-Frobenius operator is equivalent to setwise convergence of the underlying dynamic in the measure algebra. This situation is furthermore characterized by uniform mixing-like properties of the system.
Y1  - 2018
U6  - https://doi.org/10.1007/s11856-018-1671-7
SN  - 0021-2172
SN  - 1565-8511
VL  - 225
IS  - 1
SP  - 451
EP  - 463
PB  - Hebrew univ magnes press
CY  - Jerusalem
ER  - 
TY  - JOUR
A1  - Gerlach, Moritz Reinhardt
A1  - Glück, Jochen
T1  - Lower bounds and the asymptotic behaviour of positive operator semigroups
JF  - Ergodic theory and dynamical systems
N2  - If (T-t) is a semigroup of Markov operators on an L-1-space that admits a nontrivial lower bound, then a well-known theorem of Lasota and Yorke asserts that the semigroup is strongly convergent as t -> infinity. In this article we generalize and improve this result in several respects. First, we give a new and very simple proof for the fact that the same conclusion also holds if the semigroup is merely assumed to be bounded instead of Markov. As a main result, we then prove a version of this theorem for semigroups which only admit certain individual lower bounds. Moreover, we generalize a theorem of Ding on semigroups of Frobenius-Perron operators. We also demonstrate how our results can be adapted to the setting of general Banach lattices and we give some counterexamples to show optimality of our results. Our methods combine some rather concrete estimates and approximation arguments with abstract functional analytical tools. One of these tools is a theorem which relates the convergence of a time-continuous operator semigroup to the convergence of embedded discrete semigroups.
Y1  - 2017
U6  - https://doi.org/10.1017/etds.2017.9
SN  - 0143-3857
SN  - 1469-4417
VL  - 38
SP  - 3012
EP  - 3041
PB  - Cambridge Univ. Press
CY  - New York
ER  -