The first part of this work deals with open dynamical systems. A natural question of how the survival probability
depends upon a position of a hole was seemingly never addresses in the theory of open dynamical systems. We found
that this dependency could be very essential. The main results are related to the holes with equal sizes
(measure) in the phase space of strongly chaotic maps. Take in each hole a periodic point of minimal period.
Then the faster escape occurs through the hole where this minimal period assumes its maximal value. The results
are valid for all finite times (starting with the minimal period), which is unusual in dynamical systems theory
where typically statements are asymptotic when time tends to infinity. It seems obvious that the bigger the hole
is the bigger is the escape through that hole. Our results demonstrate that generally it is not true, and that
specific features of the dynamics may play a role comparable to the size of the hole.
In the second part we consider some classes of cellular automata called Deterministic Walks in Random
Environments on Z^1. At first we deal with the system with constant rigidity and Markovian distribution
of scatterers on Z^1. It is shown that these systems have essentially the same properties as DWRE on
Z^1 with constant rigidity and independently distributed scatterers. Lastly, we consider a system with
non-constant rigidity (so called process of aging) and independent distribution of scatterers. Asymptotic laws
for the dynamics of perturbations propagating in such environments with aging are obtained.
| Identifer | oai:union.ndltd.org:GATECH/oai:smartech.gatech.edu:1853/26549 |
| Date | 11 November 2008 |
| Creators | Yurchenko, Aleksey |
| Publisher | Georgia Institute of Technology |
| Source Sets | Georgia Tech Electronic Thesis and Dissertation Archive |
| Detected Language | English |
| Type | Dissertation |
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