I propose to study properties of sequences of iterations of a given continuous mapping on a compact metrizable space.
From a topological version of the Rosenthal dichotomy, the pointwise closure of the sequence either consists of the pointwise limits of subsequences and then it is called a Rosenthal compact space, or there exists a "chaotic" subsequence, where chaos is manifested by the fact the pointwise closure of the subsequence is homeomorphic to a copy of the Cech-Stone compactification of the natural numbers.
I propose to study the following two basic problems and their generalizations:
- for a given continuous mapping f, identify in terms of dynamical properties of f, when the closure of iterations of f is a Rosenthal compact space.
- construct a continuous mapping f, such that the closure of iterations of f is a complicated Rosenthal compact space in terms of ordinal ranks on the class of Rosenthal compact spaces.
Call for proposal
See other projects for this call