|Course Archives Theoretical Statistics and Mathematics Unit|
Course: Ergodic Theory
Time: Currently not offered
| Syllabus |
a) Measure preserving systems; examples. Hamiltonian dynamics and Liouville's theorem, Bernoulli shifts, Markov shifts, Rotations of the circle, Rotations of the torus, Automorphisms of the Torus, Gauss transformations, Skewproduct.
b) Poincare Recurrence lemma. Induced transformation: Kakutani towers: Rokhlins lemma. Recurrence in Topological Dynamics, Birkhoff's Recurrence theorem.
c) Ergodicity,Weak-mixing and strong-mixing and their characterisations. Ergodic theorems of Birkho and Von Neumann. Consequences of the ergodic theorems. Invariant measures on compact systems. Unique ergodicity and equidistribution. Weyl's theorem.
d) Isomorphism problem; conjugacy, spectral equivalence.
e) Transformations with discrete spectrum, Halmosvon Neumann theorem.
f) Entropy. The Kolmogorov-Sinai theorem. Calculation of Entropy. Shannon- McMillan- Breiman Theorem.
g) Flows. Birkho's Ergodic Theorem and Wiener's Ergodic Theorem for fl ows. Flows built under a function.
Suggested Texts :
1. Peter Walters, An introduction to Ergodic Theory, GTM (79), Springer (Indian reprint 2005).
2. Patrick Billingsley, Ergodic theory and information, Robert E. Krieger Publishing Co. (1978).
3. M. G. Nadkarni, Basic ergodic theory, TRIM 6, Hindustan Book Agency (1995).
4. H. Furstenberg, Recurrence in ergodic theory and combinatorial number theory, Princeton University Press (1981).
5. K. Petersen, Ergodic theory, Cambridge Studies in Advanced Mathematics (2), Cambridge University Press (1989).
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