scholarly journals Krieger's finite generator theorem for actions of countable groups Ⅱ

2019 ◽  
Vol 0 (0) ◽  
pp. 0-0
Author(s):  
Brandon Seward ◽  
Keyword(s):  
Finite Generator ◽  
Generator Theorem

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 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 On the countable generator theorem

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

The Equivalent Generator Theorem

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

scholarly journals Semi-simple locally compact monothetic semi-algebras

1969 ◽  
Vol 16 (3) ◽  
pp. 215-219 ◽  
Author(s):  
M. A. Kaashoek ◽  
T. T. West
Keyword(s):  
Spectral Properties ◽  
Locally Compact ◽  
Local Compactness ◽  
Generator Theorem ◽  
Complete Characterisation

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).

scholarly journals The asymptotic shape theorem for the frog model on finitely generated abelian groups

2021 ◽  
Author(s):  
Cristian Favio Coletti ◽  
Lucas Roberto de Lima
Keyword(s):  
Discrete Time ◽  
Polynomial Growth ◽  
Particle System ◽  
Simple Random Walk ◽  
Graphs Of Groups ◽  
Asymptotic Shape ◽  
Interacting Particle ◽  
Finite Generator ◽  
Abelian Case ◽  
Frog Model

We study the frog model on Cayley graphs of groups with polynomial growth rate $D \geq 3$. The frog model is an interacting particle system in discrete time. We consider that the process begins with a particle at each vertex of the graph and only one of these particles is active when the process begins. Each activated particle performs a simple random walk in discrete time activating the inactive particles in the visited vertices. We prove that the activation time of particles grows at least linearly and we show that in the abelian case with any finite generator set the set of activated sites has a limiting shape.

Entropy and finite generators for locally compact subshifts

1997 ◽  
Vol 17 (2) ◽  
pp. 349-368 ◽  
Author(s):  
DORIS FIEBIG ◽  
ULF-RAINER FIEBIG
Keyword(s):  
Finite Type ◽  
Embedding Theorem ◽  
Locally Compact ◽  
Bounded Degree ◽  
Shift Of Finite Type ◽  
Finite Generator ◽  
Transition Entropy

We introduce transition entropy and periodic entropy for locally compact subshifts. Finiteness of both characterizes the existence of a finite generator. Finiteness of the transition entropy characterizes the existence of a generator with bounded degree. We extend Krieger's embedding theorem and characterize the locally compact non-dense subsystems of a (compact) mixing shift of finite type.

scholarly journals PROJECTIVE LOOPS GENERATE RATIONAL LOOP GROUPS

2020 ◽  
pp. 1-27
Author(s):  
Gang Wang ◽  
Oliver Goertsches ◽  
Erxiao Wang
Keyword(s):  
Loop Group ◽  
Unitary Groups ◽  
Bäcklund Transformations ◽  
Loop Groups ◽  
Backlund Transformations ◽  
Generator Theorem ◽  
The Sacrifice ◽  
Physical Construction ◽  
Classical Construction

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.

A generator theorem for flows

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