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Geometry and Dynamics of Admissible Metrics in Measure Spaces

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A.Vershik, F.Petrov, P.Zatitskiy

Dynamical Systems · Submitted 6 May 2012

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim The main result of this paper is a new discreteness criterion for the spectrum of an ergodic transformation: we prove that the spectrum is discrete if and only if the "-entropy of the averages of some (and hence any) admissible metric over fragments of its trajectory is uniformly bounded.

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arXiv ID1205.1174 (version 3)
Pith ID6TZQLKGP, Pith's permanent record of this paper and its review
CitationA.Vershik, F.Petrov, P.Zatitskiy. "Geometry and Dynamics of Admissible Metrics in Measure Spaces." arXiv:1205.1174 (2012). https://pith.science/paper/6TZQLKGP
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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,761 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.07 · states limitation 0.07 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.20 · compares with prior results 0.07 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; .

We study a wide class of metrics in a Lebesgue space with a standard measure, the class of so-called admissible metrics. We consider the cone of admissible metrics, introduce a special norm in it, prove compactness criteria, define the "-entropy of a measure space with an admissible metric, etc. These notions and related results are applied to the theory of transformations with invariant measure; namely, we study the asymptotic properties of orbits in the cone of admissible metrics with respect to a given transformation or a group of transformations. The main result of this paper is a new discreteness criterion for the spectrum of an ergodic transformation: we prove that the spectrum is discrete if and only if the "-entropy of the averages of some (and hence any) admissible metric over fragments of its trajectory is uniformly bounded.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We study a wide class of metrics in a Lebesgue space with a standard measure, the class of so-called admissible metrics. contribution 0.85 0.90 0.01 asserted 0.95
s1 We consider the cone of admissible metrics, introduce a special norm in it, prove compactness criteria, define the "-entropy of a measure space with an admissible metric, etc. contribution 0.94 0.94 0.02 asserted 0.99
s2 These notions and related results are applied to the theory of transformations with invariant measure; namely, we study the asymptotic properties of orbits in the cone of admissible metrics with respect to a given transformation or a group of transformations. contribution 0.90 0.86 0.03 asserted 0.98
s3 The main result of this paper is a new discreteness criterion for the spectrum of an ergodic transformation: we prove that the spectrum is discrete if and only if the "-entropy of the averages of some (and hence any) admissible metric over fragments of its trajectory is uniformly bounded. result 0.96 0.96 0.99 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

We study a wide class of metrics in a Lebesgue space with a standard measure, the class of so-called admissible metrics. We consider the cone of admissible metrics, introduce a special norm in it, prove compactness criteria, define the "-entropy of a measure space with an admissible metric, etc. These notions and related results are applied to the theory of transformations with invariant measure; namely, we study the asymptotic properties of orbits in the cone of admissible metrics with respect to a given transformation or a group of transformations. The main result of this paper is a new discreteness criterion for the spectrum of an ergodic transformation: we prove that the spectrum is discrete if and only if the "-entropy of the averages of some (and hence any) admissible metric over fragments of its trajectory is uniformly bounded.

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