Spectral theory of dynamical systems

Kanigowski, Adam; Lemańczyk, Mariusz

doi:10.48550/arxiv.2006.11616

Cited by 1 publication

(1 citation statement)

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“…Remark 2.4. Our use of the term "Koopman representation" generalizes the use of the same term in [12], though technically the Koopman representation defined there is more analogous to our false Koopman representation, since there they define the Koopman representation on L 2 (Ω, µ)…”

Section: Definition 24mentioning

confidence: 99%

Noncommutative Ergodic Optimization and Unique Ergodicity

Young¹

2021

Preprint

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We extend the theory of ergodic optimization and maximizing measures to the noncommutative field of C*-dynamical systems. We then develop a commutative invariant of W*dynamical systems called an erasure in order to prove a non-commutative analog of the Jewett-Krieger Theorem. Finally, we employ this ergodic optimization machinery to provide an alternate characterization of unique erogdicity of C*-dynamical systems when the resident group action satisfies certain Choquet-theoretic assumptions.One of the guiding questions of the field of ergodic optimization is the following: Given a topological dynamical system (X, G, U), and a real-valued continuous function f ∈ C(X), what values can f dµ take when µ is an invariant Borel probability measure on X, and in particular, what are the extreme values it can take? In a joint work with I. Assani [2, Section 3], we noticed that the field of ergodic optimization was relevant to the study of certain differentiation problems dubbed spatial-temporal differentiation problems. Hoping to extend these tools to the study of spatial-temporal differentiation problems in the setting of operator-algebraic dynamical systems, this paper develops an operator-algebraic formalization of this question of ergodic optimization, re-interpreting it as a question about the values of invariant states on a C*-dynamical system. This framework is then applied to provide a characterization of certain uniquely ergodic C*dynamical systems with respect to ergodic optimizations. We also develop a novel machinery we call "erasure" in order to prove an operator-algebraic analogue to the Jewett-Krieger Theorem of classical ergodic theory.Section 1 develops the theory of ergodic optimization in the context of C*-dynamical systems, where the role of "maximizing measures" is instead played by invariant states on a C*-algebra.The framework we adopt is in fact somewhat more general than the classical framework of maximizing measures, even in the case where the underlying C*-algebra is commutative; however, the classical theory of ergodic optimization is still contained as a kind of privileged case of our framework. We also demonstrate that some of the basic results of that classical theory of ergodic optimization extend to the C*-dynamical setting.

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Section: Definition 24mentioning

confidence: 99%

Noncommutative Ergodic Optimization and Unique Ergodicity

Young¹

2021

Preprint

View full text Add to dashboard Cite

show abstract

Spectral theory of dynamical systems

Cited by 1 publication

References 219 publications

Noncommutative Ergodic Optimization and Unique Ergodicity

Noncommutative Ergodic Optimization and Unique Ergodicity

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