Transformations T:[0,1] → [0,1] with two monotonic pieces are considered. Under the assumption that T is topologically transitive and , it is proved that the invariant measures concentrated on periodic orbits are dense in the set of all invariant probability measures.
We study the class of descriptive compact spaces, the Banach spaces generated by descriptive compact subsets and their relation to renorming problems.
We relate some subsets of the product of nonseparable Luzin (e.g., completely metrizable) spaces to subsets of in a way which allows to deduce descriptive properties of from corresponding theorems on . As consequences we prove a nonseparable version of Kondô’s uniformization theorem and results on sets of points in with particular properties of fibres of a mapping . Using these, we get descriptions of bimeasurable mappings between nonseparable Luzin spaces in terms of fibres.
We study the descriptive set theoretical complexity of various randomness notions.
This paper was extensively circulated in manuscript form beginning in the Summer of 1989. It is being published here for the first time in its original form except for minor corrections, updated references and some concluding comments.
We use one-dimensional techniques to characterize the Devil’s staircase route to chaos in a relaxation oscillator of the van der Pol type with periodic forcing term. In particular, by using symbolic dynamics, we give the behaviour for certain range of parameter values of a Cantor set of solutions having a certain rotation set associated to a rational number. Finally, we explain the phenomena observed experimentally in the system by Kennedy, Krieg and Chua (in [10]) related with the appearance of...
For a space X and a regular uncountable cardinal κ ≤ |X| we say that κ ∈ D(X) if for each with |T| = κ, there is an open neighborhood W of Δ(X) such that |T - W| = κ. If then we say that X has a small diagonal, and if every regular uncountable κ ≤ |X| belongs to D(X) then we say that X has an H-diagonal. In this paper we investigate the interplay between D(X) and topological properties of X in the category of generalized ordered spaces. We obtain cardinal invariant theorems and metrization theorems...
Let f: X→ X be a topologically transitive continuous map of a compact metric space X. We investigate whether f can have the following stronger properties: (i) for each m ∈ ℕ, is transitive, (ii) for each m ∈ ℕ, there exists x ∈ X such that the diagonal m-tuple (x,x,...,x) has a dense orbit in under the action of . We show that (i), (ii) and weak mixing are equivalent for minimal homeomorphisms, that all mixing interval maps satisfy (ii), and that there are mixing subshifts not satisfying (ii)....