On some one-sided dynamics of cellular automata

A dynamical system consists of a space of all possible world states and a transformation of said space. Cellular automata are dynamical systems where the space is a set of one- or two-way infinite symbol sequences and the transformation is defined by a homogenous local rule. In the setting of cellular automata, the geometry of the underlying space allows one to define one-sided variants of some dynamical properties; this thesis considers some such one-sided dynamics of cellular automata. One main topic are the dynamical concepts of expansivity and that of pseudo-orbit tracing property. Expansivity is a strong form of sensitivity to the initial conditions while pseudo-orbit tracing property is a type of approximability. For cellular automata we define one-sided variants of both of these concepts. We give some examples of cellular automata with these properties and prove, for example, that right-expansive cellular automata are chain-mixing. We also show that left-sided pseudo-orbit tracing property together with right-sided expansivity imply that a cellular automaton has the pseudo-orbit tracing property. Another main topic is conjugacy. Two dynamical systems are conjugate if, in a dynamical sense, they are the same system. We show that for one-sided cellular automata conjugacy is undecidable. In fact the result is stronger and shows that the relations of being a factor or a susbsystem are undecidable, too

Markovian properties of continuous group actions: algebraic actions, entropy and the homoclinic group

We provide a unifying approach which links results on algebraic actions by Lind and Schmidt, Chung and Li, and a topological result by Meyerovitch that relates entropy to the set of asymptotic pairs. In order to do this we introduce a series of Markovian properties and, under the assumption that they are satisfied, we prove several results that relate topological entropy and asymptotic pairs (the homoclinic group in the algebraic case). As new applications of our method, we give a characterization of the homoclinic group of any finitely presented expansive algebraic action of (1) any elementary amenable group with an upper bound on the orders of finite subgroups or (2) any left orderable amenable group, using the language of independence entropy pairs.Comment: Minor changes. 37 pages. Adv. Math. to appea

On coarse graining and other fine problems

We study coarse grainings --- reductions of a dynamical system to its factor systems. In the literature, different variations of this problem are known under various denominations; including lumping, model reduction, aggregating, semi-conjugacy, etc. In the first half we investigate the problem of simplifying a dynamical system by reducing the number of variables and give an algorithm achieving this in some special cases. Building on the known results we extend the theory of aggregations of heuristics. We then turn to a probabilistic generalisation of these models and show that in certain cases they coarse grain onto the well-known probabilistic game of Gambler's ruin for which we prove some new results. In the second half coarse graining is used to motivate questions in topological dynamics. Given a system (X,T) we study the induced system (2$^X$,2$^T$) on the hyperspace of compact non-empty subsets of X and its periodic points. Related to this we construct an almost totally minimal system on the Cantor set. We also give a solution to a certain problem in topological dynamics related to ω-limit sets and show how a known result on the Cantor set dynamics can be seen as a consequence of a structural result about shift spaces

Book of Abstracts

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Proceedings of the First International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics

1st International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics, Kruger Park, 8-10 April 2002.This lecture is a principle-based review of a growing body of fundamental work stimulated by multiple opportunities to optimize geometric form (shape, structure, configuration, rhythm, topology, architecture, geography) in systems for heat and fluid flow. Currents flow against resistances, and by generating entropy (irreversibility) they force the system global performance to levels lower than the theoretical limit. The system design is destined to remain imperfect because of constraints (finite sizes, costs, times). Improvements can be achieved by properly balancing the resistances, i.e., by spreading the imperfections through the system. Optimal spreading means to endow the system with geometric form. The system construction springs out of the constrained maximization of global performance. This 'constructal' design principle is reviewed by highlighting applications from heat transfer engineering. Several examples illustrate the optimized internal structure of convection cooled packages of electronics. The origin of optimal geometric features lies in the global effort to use every volume element to the maximum, i.e., to pack the element not only with the most heat generating components, but also with the most flow, in such a way that every fluid packet is effectively engaged in cooling. In flows that connect a point to a volume or an area, the resulting structure is a tree with high conductivity branches and low-conductivity interstices.tm201