## Final Report Summary - TOPDS (Topological dynamics and chaos on compact metric spaces)

The main results obtained by the researcher during duration of the project (24 months) can be summarised as follows:

1. We have proved that in the case of semi-conjugacy h with shift map possessing specification property such that there is a point covered by h at most 2-to-1 (other fibers can have arbitrarily high cardinality), it is possible to transfer a distributionally scrambled set from factor (in this case, shift with specification) to extension. We have also applied this technique (together with method of isolation segments) in the investigation of dynamics of non-autonomous differential equations.

2. We have proved that every topologically mixing map with at least one fixed point contains at least one invariant (but not closed) e-scrambled set. Additionally we proved that this condition can be weakened in the case of symbolic dynamics, e.g. mixing can be replaced by transitivity.

3. We developed a formal method of measurement of complexity of nonautonomous differential equations. To do so, we introduced mathematically strict definition of topological entropy in this context and provided tools for estimation of its value (its upper or lower bounds) in terms of Poincare sections. By this tool, previous intuitive investigations of these equations became formal. Our work also provides a strict edge between chaotic and non-chaotic dynamics in this context (previous tools were working well only for nonautonomus time-periodic differential equations).

4. We used the definition of (F,G)-chaos introduced recently by Tan and Xiong together with properties of residual relations as a tool in construction of various kinds of scrambled sets. In particular, we showed that a continuous map acting on a compact metric space has an e-scrambled set if and only if it has a distributionally e-scrambled set with respect to a sequence. We also provided an example of topologically mixing map with positive topological entropy but without DC1 pairs.

5. We provided sufficient conditions for weak product recurrence expressed in terms of weakly mixing sets. In particular, our conditions work well for totally transitive maps with dense periodic points, while conditions known from the literature were implying that the map is at least topologically mixing. We also relate weak product recurrence to the problem of disjointness of dynamical systems.

By our method some insight into the structure of maps with weakly product recurrent points is obtained and in a large class of transitive systems this points are successfully localised (e.g. in any totally transitive system with dense periodic points every point with dense orbit is weakly product recurrent but not product recurrent). As a consequence of our work, a step towards the full characterisation of the class of systems disjoint from any minimal system is made.

6. We have provided a method of constructing continuous maps f:[0, 1] -> [0, 1] such that f is topologically mixing, has the shadowing property, and the inverse limit of copies of [0, 1] with f as the bonding map is the pseudoarc. Such a map can be obtained as an arbitrarily small perturbation of any topologically exact map on [0, 1]. We have therefore answered, in the affirmative, a question posed by Chen and Li in 1993.

7. We have obtained two elementary proofs showing that:

(i) transitivity and sensitivity implies dense periodicity for maps on topological graphs;

(ii) total transitivity and dense periodicity implies mixing for maps on spaces with an open subset homeomorphic with (0, 1).

As corollaries one gets new and simple proofs that Auslander-Yorke chaos implies Devaney chaos, and weak mixing implies mixing for graph maps. Although we have increased the generality, proofs are shorter than the ones existing in the literature.

1. We have proved that in the case of semi-conjugacy h with shift map possessing specification property such that there is a point covered by h at most 2-to-1 (other fibers can have arbitrarily high cardinality), it is possible to transfer a distributionally scrambled set from factor (in this case, shift with specification) to extension. We have also applied this technique (together with method of isolation segments) in the investigation of dynamics of non-autonomous differential equations.

2. We have proved that every topologically mixing map with at least one fixed point contains at least one invariant (but not closed) e-scrambled set. Additionally we proved that this condition can be weakened in the case of symbolic dynamics, e.g. mixing can be replaced by transitivity.

3. We developed a formal method of measurement of complexity of nonautonomous differential equations. To do so, we introduced mathematically strict definition of topological entropy in this context and provided tools for estimation of its value (its upper or lower bounds) in terms of Poincare sections. By this tool, previous intuitive investigations of these equations became formal. Our work also provides a strict edge between chaotic and non-chaotic dynamics in this context (previous tools were working well only for nonautonomus time-periodic differential equations).

4. We used the definition of (F,G)-chaos introduced recently by Tan and Xiong together with properties of residual relations as a tool in construction of various kinds of scrambled sets. In particular, we showed that a continuous map acting on a compact metric space has an e-scrambled set if and only if it has a distributionally e-scrambled set with respect to a sequence. We also provided an example of topologically mixing map with positive topological entropy but without DC1 pairs.

5. We provided sufficient conditions for weak product recurrence expressed in terms of weakly mixing sets. In particular, our conditions work well for totally transitive maps with dense periodic points, while conditions known from the literature were implying that the map is at least topologically mixing. We also relate weak product recurrence to the problem of disjointness of dynamical systems.

By our method some insight into the structure of maps with weakly product recurrent points is obtained and in a large class of transitive systems this points are successfully localised (e.g. in any totally transitive system with dense periodic points every point with dense orbit is weakly product recurrent but not product recurrent). As a consequence of our work, a step towards the full characterisation of the class of systems disjoint from any minimal system is made.

6. We have provided a method of constructing continuous maps f:[0, 1] -> [0, 1] such that f is topologically mixing, has the shadowing property, and the inverse limit of copies of [0, 1] with f as the bonding map is the pseudoarc. Such a map can be obtained as an arbitrarily small perturbation of any topologically exact map on [0, 1]. We have therefore answered, in the affirmative, a question posed by Chen and Li in 1993.

7. We have obtained two elementary proofs showing that:

(i) transitivity and sensitivity implies dense periodicity for maps on topological graphs;

(ii) total transitivity and dense periodicity implies mixing for maps on spaces with an open subset homeomorphic with (0, 1).

As corollaries one gets new and simple proofs that Auslander-Yorke chaos implies Devaney chaos, and weak mixing implies mixing for graph maps. Although we have increased the generality, proofs are shorter than the ones existing in the literature.