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show/hide this revision's text 2 Corrected links - I hope this time I made it

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 - link textKatok-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.

A good place to start would be - link textEinsiedler,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 - link textLindenstrauss 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 - link textDave Morris' book about Ratner's theorems.

For those who are interested in Margulis works (Arithmeticity and such), I know only of two references - link textMargulis-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 - link textErgodic Theory and Semisimple Groups.Dave Morris has a draft of book about Arithmetic groups which might be of interest as well - link textMorris - Introduction to Arithmetic Groups.
show/hide this revision's text 1 [made Community Wiki]

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 - link text, 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 - link text, 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 - link text, 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 - link text.

For those who are interested in Margulis works (Arithmeticity and such), I know only of two references - link text 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 - link text. Dave Morris has a draft of book about Arithmetic groups which might be of interest as well - link text.

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