@article{Gerlach2018,
  author    = {Gerlach, Moritz Reinhardt},
  title     = {Convergence of dynamics and the Perron-Frobenius operator},
  series = {Israel Journal of Mathematics},
  volume    = {225},
  journal   = {Israel Journal of Mathematics},
  number    = {1},
  publisher = {Hebrew univ magnes press},
  address   = {Jerusalem},
  issn      = {0021-2172},
  doi       = {10.1007/s11856-018-1671-7},
  pages     = {451 -- 463},
  year      = {2018},
  abstract  = {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.},
  language  = {en}
}
@article{GerlachGlueck2018,
  author    = {Gerlach, Moritz Reinhardt and Gl{\"u}ck, Jochen},
  title     = {Lower bounds and the asymptotic behaviour of positive operator semigroups},
  series = {Ergodic theory and dynamical systems},
  volume    = {38},
  journal   = {Ergodic theory and dynamical systems},
  publisher = {Cambridge Univ. Press},
  address   = {New York},
  issn      = {0143-3857},
  doi       = {10.1017/etds.2017.9},
  pages     = {3012 -- 3041},
  year      = {2018},
  abstract  = {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.},
  language  = {en}
}