generator theorem
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2020 ◽  
pp. 1-37 ◽  
Author(s):  
ANDREI ALPEEV ◽  
BRANDON SEWARD

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.



Author(s):  
Gang Wang ◽  
Oliver Goertsches ◽  
Erxiao Wang

We generalize Uhlenbeck’s generator theorem of ${\mathcal{L}}^{-}\operatorname{U}_{n}$ to the full rational loop group ${\mathcal{L}}^{-}\operatorname{GL}_{n}\mathbb{C}$ and its subgroups ${\mathcal{L}}^{-}\operatorname{GL}_{n}\mathbb{R}$ , ${\mathcal{L}}^{-}\operatorname{U}_{p,q}$ : they are all generated by just simple projective loops. Recall that Terng–Uhlenbeck studied the dressing actions of such projective loops as generalized Bäcklund transformations for integrable systems. Our result makes a nice supplement: any rational dressing is the composition of these Bäcklund transformations. This conclusion is surprising in the sense that Lie theory suggests the indispensable role of nilpotent loops in the case of noncompact reality conditions, and nilpotent dressings appear quite complicated and mysterious. The sacrifice is to introduce some extra fake singularities. So we also propose a set of generators if fake singularities are forbidden. A very geometric and physical construction of $\operatorname{U}_{p,q}$ is obtained as a by-product, generalizing the classical construction of unitary groups.



2019 ◽  
Vol 69 (2) ◽  
pp. 267-274
Author(s):  
Giuseppina Barbieri ◽  
Giacomo Lenzi

Abstract We give examples showing that the Kolmogorov-Sinai entropy generator theorem is false for both upper and lower Riesz entropy of MV-algebraic dynamical systems, both two sided (i.e., analogous to two sided Bernoulli shifts) and one sided (i.e., analogous to one sided Bernoulli shifts).



2019 ◽  
Vol 0 (0) ◽  
pp. 0-0
Author(s):  
Brandon Seward ◽  


2018 ◽  
Vol 215 (1) ◽  
pp. 265-310 ◽  
Author(s):  
Brandon Seward


2018 ◽  
Vol 28 (01) ◽  
pp. 1850002
Author(s):  
Ignacio S. Gomez

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.



1998 ◽  
Vol 157 (2) ◽  
pp. 255-259
Author(s):  
Michael S. Keane ◽  
Jacek Serafin
Keyword(s):  


1974 ◽  
Vol 5 (1) ◽  
pp. 45-50 ◽  
Author(s):  
Ernst Eberlein
Keyword(s):  


1969 ◽  
Vol 16 (3) ◽  
pp. 215-219 ◽  
Author(s):  
M. A. Kaashoek ◽  
T. T. West

Bonsall and Tomiuk have shown, in (3), the connection between the local compactness of a monothetic semi-algebra and the spectral properties of a generating element. This theme was developed, in (4), to give a complete characterisation of prime, strict locally compact monothetic semi-algebras in terms of the spectrum of a generator (Theorem A). Here we extend this result to the case of a semi-simple locally compact monothetic semi-algebra (Theorem B).



1930 ◽  
Vol 18 (2) ◽  
pp. 294-297 ◽  
Author(s):  
V.D. Landon
Keyword(s):  


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