Ratner's Theorems on Unipotent Flows

By Dave Witte Morris. 2003

146 citations recorded by the source corpus.

The theorems of Berkeley mathematician, Marina Ratner have guided key advances in the understanding of dynamical systems. Unipotent flows are well-behaved dynamical systems, and Ratner has shown that the closure of every orbit for such a flow is of a simple algebraic or geometric form. In "Ratner's Theorems on Unipotent Flows", Dave Witte Morris provides both an elementary introduction to these theorems and an account of the proof of Ratner's measure classification theorem. A collection of lecture notes aimed at graduate students, the first four chapters of "Ratner's Theorems on Unipotent Flows" can be read independently. The first chapter, intended for a fairly general audience, provides an introduction with examples that illustrate the theorems, some of their applications, and the main ideas involved in the proof. In the following chapters, Morris introduces entropy, ergodic theory, and the theory of algebraic groups. The book concludes with a proof of the measure-theoretic version of Ratner's Theorem. With new material that has never before been published in book form, "Ratner's Theorems on Unipotent Flows" helps bring these important theorems to a broader mathematical readership.

Subfields: Applied Mathematics, Mathematical Physics

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Citations in the corpus

LIMITING DISTRIBUTIONS OF THE CLASSICAL ERROR TERMS OF PRIME NUMBER THEORY (2013, background): “1–16]), and is also a special case of Ratner’s theorem on unipotent flows (see [33]).”
A global Torelli theorem for singular symplectic varieties (2016, background): “13] (see also the wall divisors of [Mo15]) whose definition we now recall.”
Invariant measures and the set of exceptions to Littlewood's conjecture (2006, methodology): “Ratner in a landmark series of papers ([41], [42] and others; see also the expository papers [40], [43], and the book [28]) which led to numerous applications; in particular, we use Ratner’s work heavily in this paper.”
Modified Schmidt games and a conjecture of Margulis (2010, result): “One of the major developments in the theory of homogeneous flows is a series of celebrated results of Ratner (see the surveys [Rt, KSS] and a book [Mo]) verifying conjectures of Raghunatan, Dani and Margulis on orbit closures and invariant measures of unipotent flows.”
DIRICHLET'S THEOREM ON DIOPHANTINE APPROXIMATION AND HOMOGENEOUS FLOWS (2006, background): “Ratner (see [Mo] or [KSS, Chapter 3] for a detailed account) classifies measures on G/Γ invariant under unipotent flows(2).”
Invariant and stationary measures for the SL(2,R) action on Moduli space (2013, methodology): “(For an introduction to these ideas, and also to the proof by Margulis and Tomanov [MaT] see the book [Mor].”
Invariant measures and arithmetic quantum unique ergodicity (2006, background): “Ratner in her study of horocycle flows and in her proof of Raghunathan’s conjectures, particularly [Ra82], [Ra83]; see also [Mor05], particularly §1.”
DISTRIBUTION MODULO ONE AND RATNER’S THEOREM (2006, background): “An excellent introduction to Ratner’s theory is Dave W. Morris’ recent textbook [ 11 ].”
The quantitative behaviour of polynomial orbits on nilmanifolds (2007, background): “Shah [29] obtained a similar result for arbitrary discrete unipotent (but linear) flows on a finite volume homogeneous space; the case of continuous unipotent linear flows was treated earlier by Ratner [28] (see [5] for further discussion).”
Around groups in Hilbert Geometry (2013, background): “3 of the book [69] of Witte Morris or [70] Chapter 3, Part 2).”
Morrison-Kawamata cone conjecture for hyperkahler manifolds (2014, background): “Therefore all the lifting are orbits of a Lie subgroupH1 ⊂ G (this one-parameter subgroup, isomorphic to SO(1, 1), can be identified with the group of diagonal two-by-two matrices with e and e on the diagonal under an isomorphism between SO(1, 2) and PSL(2,R), see the first chapter of Morris’ book [”