scholarly journals Krieger’s finite generator theorem for actions of countable groups III

2020 ◽  
pp. 1-37 ◽  
Author(s):  
ANDREI ALPEEV ◽  
BRANDON SEWARD
Keyword(s):  
Probability Measure ◽  
Inverse Limits ◽  
Finite Generator ◽  
Generator Theorem ◽  
Sofic Entropy ◽  
The Relationship ◽  
Invent Math

Abstract We continue the study of Rokhlin entropy, an isomorphism invariant for probability-measure-preserving (p.m.p.) actions of countablegroups introduced in Part I [B. Seward. Krieger’s finite generator theorem for actions of countable groups I. Invent. Math. 215(1) (2019), 265–310]. In this paper we prove a non-ergodic finite generator theorem and use it to establish sub-additivity and semicontinuity properties of Rokhlin entropy. We also obtain formulas for Rokhlin entropy in terms of ergodic decompositions and inverse limits. Finally, we clarify the relationship between Rokhlin entropy, sofic entropy, and classical Kolmogorov–Sinai entropy. In particular, using Rokhlin entropy we give a new proof of the fact that ergodic actions with positive sofic entropy have finite stabilizers.

scholarly journals Sofic entropy of Gaussian actions

2016 ◽  
Vol 37 (7) ◽  
pp. 2187-2222 ◽  
Author(s):  
BEN HAYES
Keyword(s):  
Probability Measure ◽  
Fourier Analysis ◽  
Harmonic Analysis ◽  
Discrete Group ◽  
Orthogonal Representation ◽  
Countable Discrete Group ◽  
Abelian Case ◽  
Sofic Entropy ◽  
Model Formalism ◽  
Invent Math

Associated to any orthogonal representation of a countable discrete group, is a probability measure-preserving action called the Gaussian action. Using the Polish model formalism we developed before, we compute the entropy (in the sense of Bowen [J. Amer. Math. Soc.23(2010) 217–245], Kerr and Li [Invent. Math.186(2011) 501–558]) of Gaussian actions when the group is sofic. Computation of entropy for Gaussian actions has only been done when the acting group is abelian and thus our results are new, even in the amenable case. Fundamental to our approach are methods of non-commutative harmonic analysis and$C^{\ast }$-algebras which replace the Fourier analysis used in the abelian case.

Slices of Hewitt–Stromberg measures and co-dimensions formula

Analysis ◽  
2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Bilel Selmi
Keyword(s):  
Probability Measure ◽  
Borel Probability Measure ◽  
Compactly Supported ◽  
Borel Probability ◽  
The Relationship

Abstract This paper studies the behavior of the lower and upper multifractal Hewitt–Stromberg functions under slices onto ( n - m ) {(n-m)} -dimensional subspaces. More precisely, we discuss the relationship between the multifractal Hewitt–Stromberg functions of a compactly supported Borel probability measure and those of slices or sections of the measure. In addition, we prove that if μ has a finite m-energy and q lies in a certain somewhat restricted interval, then these functions satisfy the expected adding of co-dimensions formula.

Invariant measures for interval maps with different one-sided critical orders

2013 ◽  
Vol 35 (3) ◽  
pp. 835-853 ◽  
Author(s):  
HONGFEI CUI ◽  
YIMING DING
Keyword(s):  
Probability Measure ◽  
Critical Points ◽  
Mild Condition ◽  
Invariant Measures ◽  
Invariant Probability Measure ◽  
Absolutely Continuous ◽  
Interval Maps ◽  
Jump Discontinuities ◽  
Point Set ◽  
Invent Math

AbstractFor an interval map whose critical point set may contain critical points with different one-sided critical orders and jump discontinuities, under a mild condition on critical orbits, we prove that it has an invariant probability measure which is absolutely continuous with respect to Lebesgue measure by using the methods of Bruin et al [Invent. Math. 172(3) (2008), 509–533], together with ideas from Nowicki and van Strien [Invent. Math. 105(1) (1991), 123–136]. We also show that it admits no wandering intervals.

scholarly journals An Estimation of the Logarithmic Timescale in Ergodic Dynamics

2018 ◽  
Vol 28 (01) ◽  
pp. 1850002
Author(s):  
Ignacio S. Gomez
Keyword(s):  
Ergodic Theory ◽  
Semiclassical Limit ◽  
Quantum Systems ◽  
Finite Generator ◽  
Generator Theorem ◽  
Recurrence Theorem

An estimation of the logarithmic timescale in quantum systems having an ergodic dynamics in the semiclassical limit, is presented. The estimation is based on an extension of the Krieger’s finite generator theorem for discretized [Formula: see text]-algebras and using the time rescaling property of the Kolmogorov–Sinai entropy. The results are in agreement with those obtained in the literature but with a simpler mathematics and within the context of the ergodic theory. Moreover, some consequences of the Poincaré’s recurrence theorem are also explored.

scholarly journals POLISH MODELS AND SOFIC ENTROPY

2016 ◽  
Vol 17 (2) ◽  
pp. 241-275 ◽  
Author(s):  
Ben Hayes
Keyword(s):  
Probability Measure ◽  
Spectral Gap ◽  
A Priori ◽  
Positive Entropy ◽  
Completely Positive ◽  
Orbit Equivalent ◽  
Factor Theorem ◽  
Sofic Entropy ◽  
Measure Preserving Actions ◽  
Theoretic Version

We deduce properties of the Koopman representation of a positive entropy probability measure-preserving action of a countable, discrete, sofic group. Our main result may be regarded as a ‘representation-theoretic’ version of Sinaǐ’s factor theorem. We show that probability measure-preserving actions with completely positive entropy of an infinite sofic group must be mixing and, if the group is nonamenable, have spectral gap. This implies that if$\unicode[STIX]{x1D6E4}$is a nonamenable group and$\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$is a probability measure-preserving action which is not strongly ergodic, then no action orbit equivalent to$\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$has completely positive entropy. Crucial to these results is a formula for entropy in the presence of a Polish, but a priori noncompact, model.

HILBERT SPACE OF ANALYTIC FUNCTIONS ASSOCIATED WITH THE MODIFIED BESSEL FUNCTION AND RELATED ORTHOGONAL POLYNOMIALS

2005 ◽  
Vol 08 (03) ◽  
pp. 505-514 ◽  
Author(s):  
NOBUHIRO ASAI
Keyword(s):  
Hilbert Space ◽  
Probability Measure ◽  
Bessel Function ◽  
Point Of View ◽  
Modified Bessel Function ◽  
Finite Moment ◽  
Pollaczek Polynomials ◽  
The Relationship ◽  
Probabilistic Point ◽  
Subordination Property

Let μ be a probability measure on I ⊂ ℝ with finite moment of all orders and γα, c1, 2 be a probability measure on ℂ which is expressed by the modified Bessel function. In this paper, we shall first consider operators B-, B+, B° and express them in terms of bosonic creation and annihilation operators. Secondly, we shall construct the Hilbert space of analytic L2 functions with respect to γα, c1, 2, [Formula: see text]. It will be seen that the Sμ-transform under certain conditions is a unitary operator from L2 (I, μ) onto [Formula: see text]. Moreover, we shall give explicit examples including Laguerre, Meixner and Meixner–Pollaczek polynomials and explain how the Hilbert space [Formula: see text] and B-, B+, B° are related from the probabilistic point of view. It will also be given the relationship between the heat kernel on ℂ and γα, c1, 2 (subordination property).

scholarly journals ON IHARA'S LEMMA FOR DEGREE ONE AND TWO COHOMOLOGY OVER IMAGINARY QUADRATIC FIELDS

2013 ◽  
Vol 09 (06) ◽  
pp. 1541-1561
Author(s):  
KRZYSZTOF KLOSIN
Keyword(s):  
Prime Power ◽  
Quadratic Field ◽  
Quadratic Fields ◽  
Cusp Forms ◽  
Imaginary Quadratic Fields ◽  
Cuspidal Cohomology ◽  
Arbitrary Weight ◽  
Relationship Of ◽  
The Relationship ◽  
Invent Math

We prove a version of Ihara's Lemma for degree q = 1, 2 cuspidal cohomology of the symmetric space attached to automorphic forms of arbitrary weight (k ≥ 2) over an imaginary quadratic field with torsion (prime power) coefficients. This extends an earlier result of the author [Ihara's lemma for imaginary quadratic fields, J. Number Theory128(8) (2008) 2251–2262] which concerned the case k = 2, q = 1. Our method is different from [Ihara's lemma for imaginary quadratic fields, J. Number Theory128(8) (2008) 2251–2262] and uses results of Diamond [Congruence primes for cusp forms of weight k ≥ 2, Astérisque196–197 (1991) 205–213] and Blasius–Franke–Grunewald [Cohomology of S-arithmetic subgroups in the number field case, Invent. Math.116(1–3) (1994) 75–93]. We discuss the relationship of our main theorem to the problem of the existence of level-raising congruences.

Interstellar gas in the central region of the Galaxy

1967 ◽  
Vol 31 ◽  
pp. 239-251 ◽  
Author(s):  
F. J. Kerr
Keyword(s):  
Central Region ◽  
Galactic Plane ◽  
Neutral Hydrogen ◽  
Continuum Emission ◽  
Galactic Centre ◽  
Interstellar Gas ◽  
Hydrogen Recombination ◽  
The Galaxy ◽  
Centre Region ◽  
The Relationship

A review is given of information on the galactic-centre region obtained from recent observations of the 21-cm line from neutral hydrogen, the 18-cm group of OH lines, a hydrogen recombination line at 6 cm wavelength, and the continuum emission from ionized hydrogen.Both inward and outward motions are important in this region, in addition to rotation. Several types of observation indicate the presence of material in features inclined to the galactic plane. The relationship between the H and OH concentrations is not yet clear, but a rough picture of the central region can be proposed.

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