Spelling suggestions: "subject:"open dynamical systems"" "subject:"ipen dynamical systems""
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Some problems in the theory of open dynamical systems and deterministic walks in random environmentsYurchenko, Aleksey 11 November 2008 (has links)
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.
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Topological and symbolic dynamics of the doubling map with a holeAlcaraz Barrera, Rafael January 2014 (has links)
This work motivates the study of open dynamical systems corresponding to the doubling map. In particular, the dynamical properties of the attractor of the doubling map when a symmetric, centred open interval is removed are studied. Using the arithmetical properties of the binary expansion of the points on the boundary of the removed interval, we study properties such as topological transitivity, the specification property and intrinsic ergodicity. The properties of the function that associates to each hole $(a,b)$ the topological entropy of the attractor of the considered dynamical system are also shown. For these purposes, a subshift corresponding to an element of the lexicographic world is associated to each attractor and the mentioned properties are studied symbolically.
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