Ergodic Theory of ℤd Actions
Proceedings of the Warwick Symposium 1993–4
Edited by Mark Pollicott
Edited by Klaus Schmidt
Publisher: Cambridge University Press
Print Publication Year: 1996
Online Publication Date:March 2010
Online ISBN:9780511662812
Paperback ISBN:9780521576888
Chapter DOI: http://dx.doi.org/10.1017/CBO9780511662812.002
Subjects: Differential and Integral Equations, Dynamical Systems and Control Theory
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Introduction.
This survey is an expanded version and elaboration of the material presented by the author at the Workshop on Algebraic and Number Theoretic Aspects of Ergodic Theory which was held in April 1994 as part of the 1993/1994 Warwick Symposium on Dynamics ofZn-actions and their connections with Commutative Algebra, Number Theory and Statistical Mechanics. The leitmotif of this paper is: Ramsey theory and ergodic theory of multiple recurrence are two beautiful, tightly intertwined and mutually perpetuating disciplines. The scope of the survey is mostly limited to Ramsey-theoretical and ergodic questions about Zn-partly because of the proclaimed goals of the Warwick Symposium and partly because of the author's hope that Zn-related combinatorics, number theory and ergodic theory can serve as an ideal lure through which the author's missionary zeal will reach as wide an audience of potential adherents to the subject as possible.
To compensate for the selective neglect of details and for the lack of full generality in some of the proofs, which were imposed by natural time and space limitations, a significant effort was spent on accentuation and motivation of ideas which lead to conjectures and techniques on which the proofs of conjectures hinge.
Here now is a brief description of the content of the five sections constituting the body of this survey. In Section 1 three main principles of Ramsey theory are introduced and their connection with the ergodic theory of multiple recurrence is emphasized.
pp. i-iv
pp. v-vi
pp. vii-viii
pp.
1 - Ergodic Ramsey Theory : Read PDF
pp. 1-62
2 - Flows on homogeneous spaces : Read PDF
pp. 63-112
3 - The variational principle for Hausdorff dimension : Read PDF
pp. 113-126
4 - Boundaries of invariant Markov Operators: The identification problem : Read PDF
pp. 127-176
5 - Squaring and cubing the circle – Rudolph's theorem : Read PDF
pp. 177-184
6 - A survey of recent K-theoretic invariants for dynamical systems : Read PDF
pp. 185-236
pp. 237-258
8 - Overlapping cylinders: the size of a dynamically defined Cantor-set : Read PDF
pp. 259-272
pp.
1 - Uniformity in the polynomial Szemérdi theorem : Read PDF
pp. 273-296
2 - Some 2-d symbolic dynamical systems: Entropy and mixing : Read PDF
pp. 297-306
3 - A note on certain rigid subshifts : Read PDF
pp. 307-318
4 - Entropy of graphs, semigroups and groups : Read PDF
pp. 319-344
5 - On representation of integers in Linear Numeration Systems : Read PDF
pp. 345-368
6 - The structure of ergodic transformations conjugate to their inverses : Read PDF
pp. 369-386
7 - Approximatiom by periodic transformations and diophantine approximation of the spectrum : Read PDF
pp. 387-402
8 - Invariant σ-algebras for ℤd-actions and their applications : Read PDF
pp. 403-414
9 - Large deviations for paths and configurations counting : Read PDF
pp. 415-432
10 - A zeta function for ℤd-actions : Read PDF
pp. 433-450
11 - The dynamical theory of tilings and Quasicrystallography : Read PDF
pp. 451-474
12 - Approximations of groups and group actions, Cayley topology : Read PDF
pp. 475-484