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The dissertation describes ergodic properties of some stochastic dynamicalsystems generated by Markov chains with values in the state space which is aPolish space. The mathematical model describing the process of cell division isanalyzed. The assumptions are satisfied, among others, by the model introducedby J.J. Tyson and K.B. Hannsgen, J. Math. Biol. (1986). Within the thesis theexistence of a unique invariant measure is established and asymptotic stabilityof the model is verified. In addition, the rate of convergence of the sequenceof measures, transformed by the subsequent iterations of Markov operator, tothe unique invariant measure is evaluated. Further investigation of the model(in terms of its ergodic properties) concerns establishing the proofs of thecentral limit theorem and the law of the iterated logarithm. The resultspresented in the dissertation solve interesting mathematical problems, but,since they are inspired by the biological process of cell division, they may bealso important for biology.

PhD dissertation (in Polish): Ergodyczne w{\l}asno\'sci pewnych stochastycznych uk{\l}ad\'ow dynamicznych