Leírás
I will present a topic on which I have written my Student Research Conferece (TDK) thesis. A class of real-to-real maps is considered, which have a countable Markov paritition and are linear on the elements of the partition. We start by considering properties of a discrete time dynamical system presented in the third problem of the Miklós Schweitzer competition 2022., which was proposed by my advisor, Zoltán Buczolich. This system has a non-compact domain and possesses a natural Markov partition, if we allow countably infinite partitions of the domain. Through this example a variety of methods can be demonstrated which are powerful tools of the analysis of dynamical systems. One such method is that we introduce a topologically conjugate symbolic dynamical system.
We will touch upon the shadowing property, which ensures that numerically simulated orbits of chaotic dynamical systems, with possible errors still preserve some information about the true orbits.
For describing the long-time behaviour of such systems, it is best to employ methods of ergodic theory. For a wide class of Markov transformations, we find the absolutely continuous invariant measures and we prove a strong statement about the convergence of pushforwards of absolutely continuous measures.
For a more restricted class of transformations the periodic orbits turn out to be the orbits of rational numbers. The analysis of these orbits involves applications of Cantor series. Characterization of such systems is quite interesting.
The zoom link to the talk is:
https://us06web.zoom.us/j/97594629945?pwd=MmFNaVk4a1FhdjEvc2RRdGdod0FpZz09 .