LISEDIDYS · Limit sets of discrete dynamical systems

Limit sets of discrete dynamical systems

In real life we often encounter problems whose evolution can be defined in fixed state space according to a fixed rule. Populations, epidemics and economies at all scales are such examples, which mathematicians are trying to model through so-called dynamical systems. In the project we are interested in discrete dynamical systems, that is systems which evolve in discrete time steps. The aim of the project is studying long time behaviour of such systems (expressed in terms of the limit sets) in the future or the past. As one of research goals, we investigate structure, topological size and other topological and dynamical properties of limit sets. We focus on low dimensional dynamical systems such as interval maps, maps acting on topological graphs, dendrites, etc. While the limit sets of forward trajectories were deeply studied in past decades, the limit sets of backward trajectories, which should be regarded as sources of trajectories, have not been that much explored so far. Our aim was obtaining good characterization of the properties of limit sets of backward trajectories for interval and more widely, graph maps or even maps acting on general topological spaces. In the context of forward dynamics, good characterizations exist on topological graphs, however dynamical properties of maps on dendrites are not completely characterized. Therefore, another goal was understanding of limit sets of forward trajectories for maps acting on dendrites and related open questions from measure theory about spectrum of invariant measures when topological entropy is zero.

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