Recurrence in Ergodic Theory and Combinatorial Number Theory

By H. Furstenberg. 2014

1,002 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/50926c2d669722cb7412c68c91d4cac5f4626138

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In Mathematical Physics

Citations in the corpus

Recurrence properties of hypercyclic operators (2014, background): “Indeed, if s 1 , s 2 ∈ N ( x, U n ), then T s 2 x ∈ U and T s 1 − s 2 + n ( T s 2 x ) = T n ( T s 1 x ) ∈ V On the other hand, we know that if A is a set with positive upper Banach density, then A − A is syndetic [12, Proposition 3.19].”
The logarithmic Sarnak conjecture for ergodic weights (2017, neutral): “2 and is motivated by analogous constructions made by Furstenberg in [27].”
Invariant measures for frequently hypercyclic operators (2014, background): “in [15]; it will also be briefly recalled at the beginning of Section 2.”
Pointwise ergodic theorems for non-conventional bilinear polynomial averages (2020, background): “idistribution theory of nilmanifolds; see for instance [3], [4], [26] for further discussion. These results are also related to recurrence and Roth and Szemeredi type theorems (see e.g., [29], [30], [31], [5], [71]) but we will not discuss these topics further here. 1.2. Linear averages. We now reca”
Recurrent Linear Operators (2013, methodology): “First we give an equivalent characterization of recurrence by means of the following well known proposition; see for example [23].”
Recurrence properties and disjointness on the induced spaces (2013, background): “s a minimal point if and only if N(x,U)is syndetic for any neighborhood U of x; a t.d.s. (X,T)is weakly mixing if and only if N(U,V) is thick for any non-empty open subsetsU,V of X (see, for example, [6, 7]); and a t.d.s. (X,T)is an E-system if and only if there is a transitive pointx ∈ X such that ”
Nil Bohr-sets and almost automorphy of higher order (2014, background): “[51, 21] We will not give the proof of this theorem since it is a special case of Theorem 8.”