  The dissertation describes ergodic properties of some stochastic dynamical
systems generated by Markov chains with values in the state space which is a
Polish space. The mathematical model describing the process of cell division is
analyzed. The assumptions are satisfied, among others, by the model introduced
by J.J. Tyson and K.B. Hannsgen, J. Math. Biol. (1986). Within the thesis the
existence of a unique invariant measure is established and asymptotic stability
of the model is verified. In addition, the rate of convergence of the sequence
of measures, transformed by the subsequent iterations of Markov operator, to
the unique invariant measure is evaluated. Further investigation of the model
(in terms of its ergodic properties) concerns establishing the proofs of the
central limit theorem and the law of the iterated logarithm. The results
presented in the dissertation solve interesting mathematical problems, but,
since they are inspired by the biological process of cell division, they may be
also important for biology.
