Dilating Covariant Representations of a Semigroup Dynamical System Arising from Number Theory

The spectral C*-algebra of the discrete product systems of H.T. Dinh is shown to be a twisted semigroup crossed product whenever the product system has a twisted unit. The covariant representations of the corresponding dynamical system are always faithful, implying the simplicity of these crossed products; an application of a recent theorem of G.J. Murphy gives their nuclearity. Furthermore, a semigroup of endomorphisms of B(H) having an intertwining projective semigroup of isometries can be extended to a group of automorphisms of a larger Type I factor.

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Dynamical system proof of Fermat’s little theorem: An alternative approach

Malaysian Journal of Fundamental and Applied Sciences ◽

10.11113/mjfas.v14n3.1019 ◽

2018 ◽

Vol 14 (3) ◽

pp. 331-333

Author(s):

Olamide Funmilayo Florence ◽

Tahir Ahmad ◽

Adaraniwon Amos Olalekan

Keyword(s):

Dynamical System ◽

Number Theory ◽

Alternative Approach ◽

Fermat’S Little Theorem

Fermat’s little theorem has been proved using different mathematical approaches, which majority of them are based on number theory. These approaches have only exposed the usability of Fermat’s little theorem for logical, linear and structural predictions. Only small numbers of attempts had only been made to proof Fermat’s little theorem from other perspectives. This paper exhibits an alternative approach to proof the Fermat’s little theorem via dynamical system. Two lemmas are proven with respect to a redefined function, Tn (x) in order to achieve the task.

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Ball-avoiding theorems

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700001000 ◽

2000 ◽

Vol 20 (6) ◽

pp. 1821-1849 ◽

Cited By ~ 2

Author(s):

DOMOKOS SZÁSZ

Keyword(s):

Dynamical System ◽

Number Theory ◽

Phase Space ◽

Hausdorff Dimension ◽

Open Subset ◽

Small Subset ◽

Interested Reader ◽

Rigorous Results ◽

Hard Ball

Consider a nice hyperbolic dynamical system (singularities not excluded). Statements about the topological smallness of the subset of orbits, which avoid an open subset of the phase space (for every moment of time, or just for a not too small subset of times), play a key role in showing hyperbolicity or ergodicity of semi-dispersive billiards, especially, of hard-ball systems. As well as surveying the characteristic results, called ball-avoiding theorems, and giving an idea of the methods of their proofs, their applications are also illustrated. Furthermore, we also discuss analogous questions (which had arisen, for instance, in number theory), when the Hausdorff dimension is taken instead of the topological one. The answers strongly depend on the notion of dimension which is used. Finally, ball-avoiding subsets are naturally related to repellers extensively studied by physicists. For the interested reader we also sketch some analytical and rigorous results about repellers and escape times.

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The Plaid Model

10.23943/princeton/9780691181387.001.0001 ◽

2019 ◽

Author(s):

Richard Evan Schwartz

Keyword(s):

Dynamical System ◽

Number Theory ◽

Continued Fractions ◽

Planetary Motion ◽

Elementary Number Theory ◽

Convex Shape ◽

Tile System ◽

Outer Billiards ◽

Combinatorial Model ◽

Self Similar

Outer billiards provides a toy model for planetary motion and exhibits intricate and mysterious behavior even for seemingly simple examples. It is a dynamical system in which a particle in the plane moves around the outside of a convex shape according to a scheme that is reminiscent of ordinary billiards. This book provides a combinatorial model for orbits of outer billiards on kites. The book relates these orbits to such topics as polytope exchange transformations, renormalization, continued fractions, corner percolation, and the Truchet tile system. The combinatorial model, called “the plaid model,” has a self-similar structure that blends geometry and elementary number theory. The results were discovered through computer experimentation and it seems that the conclusions would be extremely difficult to reach through traditional mathematics. The book includes an extensive computer program that allows readers to explore the materials interactively and each theorem is accompanied by a computer demonstration.

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