On coarse graining and other fine problems

Puljiz, Mate (2017). On coarse graining and other fine problems. University of Birmingham. Ph.D.

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Abstract

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.

Type of Work:

Thesis (Doctorates > Ph.D.)

Award Type:

Doctorates > Ph.D.

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