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Dynamics of Homeomorphisms, Noninvertible Maps and Flows with Respect to Minimality

Final Activity Report Summary - DS-MINIM (Dynamics of Homeomorphisms, Noninvertible Maps and Flows with Respect to Minimality)

The project was focused on studying dynamical systems given by an action of a continuous map or homeomophism on a metric space with respect to minimality and related properties such as transitivity and aperiodicity. We constructed a new, rich class of minimal systems with dynamics given by a continuous map, almost totally disconnected dynamical systems, with complicated behaviour; we fully described minimal sets on these spaces and proved several others general results; as a consequence we got a full topological characterization of minimal sets on dendrites.

We studied minimal sets of homeomorphisms on noncompact surfaces of finite type; we gave partial description of possible minimal sets in this setting; we constructed an example of embedded non-compact Cantor set as minimal set in this case. We studied fixed point free homeomorphisms of the open and closed annulus; we got partial results about the existence of foliations with free leaves.

We studied almost periodically forced systems; we constructed an explicit example of embedded Denjoy dynamics in a system with no invariant curves; we proved other related results in this setting. Two non-equivalent definitions of minimality are discussed in a general setting; a panorama of old and new results of general character is presented.