吉布斯度量的谱三元和迪克斯米尔痕量表示:理论与实例

IF 1.6 2区 数学 Q2 MATHEMATICS, APPLIED Nonlinearity Pub Date : 2024-09-05 DOI:10.1088/1361-6544/ad7009
L Cioletti, L Y Hataishi, A O Lopes, M Stadlbauer
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引用次数: 0

摘要

在本文中,我们研究了与膨胀和弱膨胀映射相关的谱三元组和非交换期望。为此,我们对 Perron-Frobenius-Ruelle 定理进行了推广,得到了算子的多项式衰减,从而证明了动态定义的 ζ 函数在其临界参数处的可微分性。然后,我们将夏普的谱三元构造推广到这一环境中,并在相关谱度量是非退化的、谱三元的非交换期望与热力学形式主义的相关平衡态的积分是共线的情况下提供了标准。由于我们的一般设置,我们能够同时分析流形或连通分形上的膨胀映射、有限类型的子移动以及统计物理学中的戴森模型,这凸显了非交换几何的统一性。此外,我们还推导出了与某类病态连续势相关的ζ函数的明确表示,并举例说明了通过相关zeta函数作为非交换期望的表示在哪些情况下成立,以及在哪些情况下不成立。
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Spectral triples and Dixmier trace representations of Gibbs measures: theory and examples
In this paper we study spectral triples and non-commutative expectations associated to expanding and weakly expanding maps. In order to do so, we generalise the Perron–Frobenius–Ruelle theorem and obtain a polynomial decay of the operator, which allows to prove differentiability of a dynamically defined ζ-function at its critical parameter. We then generalise Sharp’s construction of spectral triples to this setting and provide criteria when the associated spectral metric is non-degenerate and when the non-commutative expectation of the spectral triple is colinear to the integration with respect to the associated equilibrium state from thermodynamic formalism. Due to our general setting, we are able to simultaneously analyse expanding maps on manifolds or connected fractals, subshifts of finite type as well as the Dyson model from statistical physics, which underlines the unifying character of noncommutative geometry. Furthermore, we derive an explicit representation of the ζ-function associated to a particular class of pathological continuous potentials, giving rise to examples where the representation as a non-commutative expectation via the associated zeta function holds, and others where it does not hold.
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来源期刊
Nonlinearity
Nonlinearity 物理-物理:数学物理
CiteScore
3.00
自引率
5.90%
发文量
170
审稿时长
12 months
期刊介绍: Aimed primarily at mathematicians and physicists interested in research on nonlinear phenomena, the journal''s coverage ranges from proofs of important theorems to papers presenting ideas, conjectures and numerical or physical experiments of significant physical and mathematical interest. Subject coverage: The journal publishes papers on nonlinear mathematics, mathematical physics, experimental physics, theoretical physics and other areas in the sciences where nonlinear phenomena are of fundamental importance. A more detailed indication is given by the subject interests of the Editorial Board members, which are listed in every issue of the journal. Due to the broad scope of Nonlinearity, and in order to make all papers published in the journal accessible to its wide readership, authors are required to provide sufficient introductory material in their paper. This material should contain enough detail and background information to place their research into context and to make it understandable to scientists working on nonlinear phenomena. Nonlinearity is a journal of the Institute of Physics and the London Mathematical Society.
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