On Some Aspects of the Theory of Anosov Systems

By Gregory Margulis. 2004

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In Mathematical Physics

Citations in the corpus

Anosov groups: local mixing, counting and equidistribution (2020, methodology): “When Γ is a cocompact lattice in a rank one Lie group and H is compact, this goes back to Margulis’ thesis from 1970 (see [29] for an English translation published in 2004).”
Manin's and Peyre's conjectures on rational points and adelic mixing (2006, methodology): “c bounds [BS, CU, Cl1, GJ, JL, Ro]. • Using decay, deduce that #(G(K) ∩BT) ∼ vol(GHι ∩ BT) as T→ ∞. A connection between decay of matrix coeﬃcients and counting problems was ﬁrst observed by Margulis [Mar] in his thesis in 1970 and exploited further by Eskin and McMullen [EM]. Here we apply this ide”
Apollonian circle packings and closed horospheres on hyperbolic 3-manifolds (2008, background): “This approach appears already in Margulis’s 1970 thesis [25]; see also [12].”
Anosov flows and dynamical zeta functions (2012, background): “42 In fact, it is bounded by C#e htop(φ1)t and there is even an asymptotic formula, see [34], which we are improving in Theorem 2.”
Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces (2007, background): “While there is overhead in developing their basic properties, they offer a clean formalism for analysis on homogeneous spaces.”
DIRICHLET'S THEOREM ON DIOPHANTINE APPROXIMATION AND HOMOGENEOUS FLOWS (2006, methodology): “for any t ∈ R. In this case Theorem 2.2 coincides with [KM1, Proposition 2.2.1], where it was deduced from mixing of the G-action on G/Γ, using an argument dating back to the Ph.D. Thesis of Margulis [Ma2]. On the other hand, H is strictly contained in the expanding horospherical subgroup relative t”
Ratner's Theorems on Unipotent Flows (2003, background): “Dani [12, 13, 14] then proved a number of results, including the first example of an Orbit Closure Theorem for actions of non-horospherical unipotent subgroups of a semisimple Lie group — namely, for “generic” one-parameter unipotent subgroups of SL(3,R).”
Distribution of periodic torus orbits and Duke's theorem for cubic fields (2007, background): “Bowen and Margulis, independently [5, 42], proved that the set of geodesics of length ≤ L, considered as closed orbits of the geodesic flow on X, are equidistributed w.”
Equidistribution and counting for orbits of geometrically finite hyperbolic groups (2010, methodology): “This approach for counting via mixing was first used by Margulis’s 1970 thesis [24].”
Effective equidistribution of S-integral points on symmetric varieties (2007, methodology): “tions ([12], [13]) and of Jeanquartier on ﬁber integrations [30]. We remark that the approach for counting via mixing was initiated in 1970 by Margulis in his dissertation on Anosov dynamical systems [34]. Recently similar mixing properties in an adelic setting have been used in the study of rationa”