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Hi everybody! I'm interested in Lie Theory and its connections to Dynamical Systems theory. I am starting my studies and would like references to articles on the subject.

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I am also interested in this connection. Searching the internet and literature hasn't been too forthcoming yet, but I have only just begun. It seems to me that a likely connection would be through representations of amenable groups. Jaoby, have you found anything useful in this direction on your own? –  Eric A. Bunch Apr 1 '11 at 23:47
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Also, this question should probably be made community wiki. –  Eric A. Bunch Apr 1 '11 at 23:48
    
Have you looked into Renato Feres's book Dynamical Systems and Semisimple groups? I found it beautifully written and very accessible. You should find many pointers to the literature in it. –  Theo Buehler Apr 2 '11 at 4:53
    
Take a look at the Handbook of Dynamical Systems. –  Steve Huntsman Apr 2 '11 at 15:32
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2 Answers 2

up vote 7 down vote accepted

The connections between Dynamics and Lie Groups (or Algebraic groups) comes mainly in two flavours: 1. Smooth dynamics, like others have stated Hamiltonian dyanmics and differntial equations. 2. Applications of Ergodic theory and Topological dynamics to Lie groups (or more generally, homogenuous spaces), or as Lindenstrauss' calls it - homogenuous dynamics.

The homogenuous dynamics realm is again divided to two main areas: 1. "Geometric Applications", i.e. most of Margulis works (rigidty and such). Those are problems that deal directly with these settings. 2. "Other applications", mainly number theortical applications, which basically can be modelled on such spaces and dynamical methods (such as orbit classification or measure classification) come to use.

I find myself more expert on 2.2 side, and I don't know anything about smooth dynamics, so I'll leave you with just one reference - Katok-Introduction to the Modern Theory of Dynamical Systems, which is some sort of general encyclopedia, and might be a good place to start your journey.

About homogenuous dynamics. The area doesn't have a usuall reference, and to be exact, there are hardly any references at all. A good place to start would be - Einsiedler,Ward - Ergodic Theory With a View Towards Number Theory, this relatively a new book in the GTM, which is well written, and give you introduction to ergodic theory, and in the later part he proves Ratner theorems for SL_2 (Furstenberg, Danni, Danni-Smilie). He also discuss some of the dynamics of nilpotent systems, such as the Heisenberg group (which is the starting point toward the Green-Tao theorem).

For the more advanced reader, the best place would be Elon's own notes - Lindenstrauss notes from a prev. course in HU, another good place would be the Clay Pisa proceedings, containing lecture notes of Eskin regarding Ratner theorems, and a paper by Lindenstrauss and Einsiedler about their work on diagonalizable actions. If one is particularly interested in Ratner theorems, one can look in - Dave Morris' book about Ratner's theorems.

For those who are interested in Margulis works (Arithmeticity and such), I know only of two references - Margulis-Discrete subgroups of semisimple Lie groups which is out of print and extremely hard to find (and also hard to understand, one needs some familiarity with Lie groups and algebraic groups), and the other one is a book by Zimmer - Ergodic Theory and Semisimple Groups. Dave Morris has a draft of book about Arithmetic groups which might be of interest as well - Morris - Introduction to Arithmetic Groups.

Well I'm hoping I gave you enough reading for some time.

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One of the most important connections of the two fields can be found in the theory of Hamiltonian dynamical systems with Lie groups being the symmetry groups. The interplay of these leads to many interesting concepts (including, inter alia, the classical R-matrix) and results. For starters you can try the books Applications of Lie Groups to Differential Equations by Olver and Integrable Systems of Classical Mechanics and Lie Algebras by Perelomov, and the survey paper Integrable Systems and Factorization Problems by Semenov-Tian-Shansky.

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