Ergodic Theory: with a view towards Number Theory

By Manfred Einsiedler, T. Ward. 2010

723 citations recorded by the source corpus.

Subfields: Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/3ada1c7200418531ef024edf674a353ea0a62516

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In Statistical and Nonlinear Physics

Citations in the corpus

Sarnak’s Conjecture: What’s New (2017, background): “It means that we assume that the reader is familiar with at least basics of ergodic theory, but often more than that is required, monographs [38, 55, 73, 75, 164] are among best sources to be consulted.”
On the Pisot Substitution Conjecture (2015, methodology): “Tilings T ∈ Ω φ can be thought of as interpolates of elements of X φ : formally, the system (Ω φ , R ) is topologically conjugate with the suspension system ( ˆ X φ , R ) of ( X φ , s ) with roof function given by (cid:96) (see, e.g. , [51, Chap.2] for a discussion of suspension).”
Stable sets and mean Li–Yorke chaos in positive entropy systems (2012, neutral): “…there exists a set X 1 ∈ B m with m ( X 1 ) = 1 such that for any x ∈ X 1 , lim n → + ¥ m n , x = m x under the weak ∗ topology (see for example [11, Corollary 5.21]), and for each n ≥ 0, m n , x ( W s ( x , T )) = 1 and supp ( m n , x ) ⊆ supp ( m n + 1 , x ) ⊆ supp ( m x ) (3.2) by Lemma 2.1…”
Diagonal actions on locally homogeneous spaces (2008, methodology): “For more details we again refer to [Wal82], [Gla03], or [EW07].”
Some open problems on multiple ergodic averages (2011, background): “[71] M.”
Badly approximable systems of affine forms, fractals, and Schmidt games (2009, methodology): “The proofs of our results require two tools: Schmidt games (see [17] for a reference) and the basic concepts concerning flows on the space of unimodular lattices (see Chapter 9 of [6] or [1] for a reference).”
Singular systems of linear forms and non-escape of mass in the space of lattices (2014, background): “For if μ is not ergodic, we can write μ as an integral of its ergodic components μ = ∫ μrdν(r) for some probability space (E, ν), see for example [EW11].”
Combinatorial Independence and Sofic Entropy (2012, background): “Introduction Among the various phenomena in dynamics associated with randomness, weak mixing and entropy stand out for the depth of their theory and the breadth of their applications (see for example [13, 15]).”
Automorphisms with quasi-discrete spectrum, multiplicative functions and average orthogonality along short intervals (2015, neutral): “observing as in [10,12] that the last coordinate of T (x1, .”
A proof of Furstenberg's conjecture on the intersections of ⨯p- and ⨯q-invariant sets (2016, background): “We refer the reader to [3, 4, 41] for more information.”
The Bouncy Particle Sampler: A Nonreversible Rejection-Free Markov Chain Monte Carlo Method (2015, methodology): “The law of large numbers then follows by Birkhoﬀ’s pointwise ergodic theorem; see e.g. [9, Theorem 2.30, Section 2.6.4].”