Topological dynamics in one-dimensional spaces

Project Details

Description

My field of research is mainly dynamics of interval maps. We worked on the problem of existence of measures with maximal entropy and we would like to strengthen some results that were proven for sufficiently smooth interval maps. For this, it is necessary to do a sharp study of the critical set. An alternative method would be to use the links between an interval map and its Markov diagram and to understand better which properties are shared by the two dynamics. Moreover, we would like to study measures with maximal entropy for interval maps that are only continuous. On the other hand, we are interested in chaos in one-dimensional systems. For interval maps, there are many relations between the properties linked with the notion of chaos. We would like to know which of them remain valid for continuous maps of one-dimensional spaces. For some of these properties, slight changes in the hypotheses, or some additional hypotheses, may be needed.

To adapt Sharkovskii's Theorem, which deals with the periods of periodic points which can coexist for an interval map, much remains to do. We would like to study the structure of periodic cycles of one-dimensional systems, which is already a research area of the Universitat Autonoma de Barcelona. A one-year stay there would allow me to work with experienced researchers who work on one-dimensional dynamics and is likely to lead to joint papers. The Universitat Autonoma de Barcelona is an important place for research in dynamics of maps of one-dimensional spaces. Having a post-doctoral position there will allow me to acquire new skills in this domain, and see these systems with a different point of view. In particular we could become more familiar with the tools that were introduced to study the structure of periodic cycles for maps of graph-like spaces ( this structure is much more complicated than for maps of the interval ) : On the other hand we have a good knowledge of the topological Markovchains.
StatusFinished
Effective start/end date1/11/0231/10/03

Funding

  • European Commission: €54,884.00

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