scholarly journals Orbit growth for algebraic flip systems

2014 ◽  
Vol 35 (8) ◽  
pp. 2613-2631 ◽  
Author(s):  
RICHARD MILES
Keyword(s):  
Abelian Group ◽  
Periodic Point ◽  
Topological Entropy ◽  
Compact Abelian Group ◽  
Dihedral Group ◽  
Structure Theorem ◽  
Point Counting ◽  
Growth Estimate ◽  
Fundamental Structure ◽  
Image Position

An algebraic flip system is an action of the infinite dihedral group by automorphisms of a compact abelian group $X$. In this paper, a fundamental structure theorem is established for irreducible algebraic flip systems, that is, systems for which the only closed invariant subgroups of $X$ are finite. Using irreducible systems as a foundation, for expansive algebraic flip systems, periodic point counting estimates are obtained that lead to the orbit growth estimate $$\begin{eqnarray}Ae^{hN}\leqslant {\it\pi}(N)\leqslant Be^{hN},\end{eqnarray}$$ where ${\it\pi}(N)$ denotes the number of orbits of length at most $N$, $A$ and $B$ are positive constants and $h$ is the topological entropy.

scholarly journals A Theorem on Algebras of Measures on Topological Groups

1959 ◽  
Vol 11 (4) ◽  
pp. 195-206 ◽  
Author(s):  
J. H. Williamson
Keyword(s):  
Banach Space ◽  
Abelian Group ◽  
Compact Abelian Group ◽  
Locally Compact Abelian Group ◽  
Topological Groups ◽  
Finite Sets ◽  
Locally Compact ◽  
Borel Measures ◽  
Bounded Complex ◽  
Image Position

Let G be a locally compact Abelian group, and the set of bounded complex (regular countably-additive Borel) measures on G. It is well known that becomes a Banach space if the norm is defined bythe supremum being over all finite sets of disjoint Borel subsets of G.

scholarly journals On uniformly distributed sequences of integers and Poincaré recurrence III

2003 ◽  
Vol 68 (2) ◽  
pp. 345-350
Author(s):  
R. Nair
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Probability Space ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Bohr Compactification ◽  
Poincare Recurrence ◽  
Uniformly Distributed Sequences ◽  
Image Position ◽  
Measure Preserving Transformations

Let S be a semigroup contained in a locally compact Abelian group G. Let Ĝ denote the Bohr compactification of G. We say that a sequence contained in S is Hartman uniform distributed on G iffor any character χ in Ĝ. Suppose that (Tg)g∈s is a semigroup of measurable measure preserving transformations of a probability space (X, β, μ) and B is an element of the σ-algebra β of positive μ measure. For a map T: X → X and a set A ⊆ X let T−1A denote {x ∈ X: Tx ∈ A}. In an earlier paper, the author showed that if k is Hartman uniform distributed thenIn this paper we show that ≥ cannot be replaced by =. A more detailed discussion of this situation ensues.

scholarly journals Multipliers for Amalgams and the Algebra S0(G)

1987 ◽  
Vol 39 (1) ◽  
pp. 123-148 ◽  
Author(s):  
Maria L. Torres De Squire
Keyword(s):  
Abelian Group ◽  
Haar Measure ◽  
Compact Abelian Group ◽  
Dual Group ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
The Difference ◽  
Image Position

Throughout the whole paper G will be a locally compact abelian group with Haar measure m and dual group Ĝ. The difference of two sets A and B will be denoted by A ∼ B, i.e.,For a function f on G and s ∊ G, the functions f′ and fs will be defined by

Gap Series on Groups and Spheres

1966 ◽  
Vol 18 ◽  
pp. 389-398 ◽  
Author(s):  
Daniel Rider
Keyword(s):  
Abelian Group ◽  
Trigonometric Polynomial ◽  
Haar Measure ◽  
Compact Abelian Group ◽  
Dual Group ◽  
Gap Series ◽  
Image Position

Let G be a compact abelian group and E a subset of its dual group Γ. A function ƒ ∈ L1(G) is called an E-function if for all γ ∉ E wheredx is the Haar measure on G. A trigonometric polynomial that is also an E-function is called an E-polynomial.

scholarly journals Inequalities related to those of Hausdorff-Young

1972 ◽  
Vol 6 (2) ◽  
pp. 185-210 ◽  
Author(s):  
R.E. Edwards
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Trigonometric Polynomials ◽  
Fourier Image ◽  
Integrable Functions ◽  
Inclusion Relations ◽  
Image Position

This note establishes the impossibility of certain inequalities of the form holding for all trigonometric polynomials f on an infinite compact abelian group G. From this is deduced the impossibility of corresponding inclusion relations of the type or where FS denotes the Fourier image of the set S of integrable functions on G.

scholarly journals Supports and singular supports of pseudomeasures

1966 ◽  
Vol 6 (1) ◽  
pp. 65-75 ◽  
Author(s):  
R. E. Edwards
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Closed Subset ◽  
Sufficient Conditions ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Singular Support ◽  
Image Position

SummaryLet G denote a Hausdorff locally compact Abelian group which is nondiscrete and second countable. The main results (Theorems (2.2) and (2.3)) assert that, for any closed subset E of G there exists a pseudomeasure s on G whose singular support is E; and that if no portion of E is a Helson set, then such an s may be chosen having its support equal to E. There follow (Corollaries (2.2.4) and (2.3.2)) sufficient conditions for the relations to hold for some pseudomeasure s, E and F being given closed subsets of G. These results are analogues and refinements of a theorem of Pollard [4] for the case G = R, which asserts the existence of a function in L∞(R) whose spectrum coincides with any preassigned closed subset of R.

scholarly journals A bridge theorem for the entropy of semigroup actions

2020 ◽  
Vol 8 (1) ◽  
pp. 46-57
Author(s):  
Anna Giordano Bruno
Keyword(s):  
Abelian Group ◽  
Topological Entropy ◽  
Compact Abelian Group ◽  
Locally Compact Abelian Group ◽  
Finitely Generated ◽  
Semigroup Action ◽  
Locally Compact ◽  
Semigroup Actions ◽  
Dual Action ◽  
Totally Disconnected

AbstractThe topological entropy of a semigroup action on a totally disconnected locally compact abelian group coincides with the algebraic entropy of the dual action. This relation holds both for the entropy relative to a net and for the receptive entropy of finitely generated monoid actions.

The uniform compactification of a locally compact abelian group

1990 ◽  
Vol 108 (3) ◽  
pp. 527-538 ◽  
Author(s):  
M. Filali
Keyword(s):  
Abelian Group ◽  
Compact Abelian Group ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Complex Functions ◽  
Discrete Semigroups ◽  
Continuous Complex ◽  
Image Position ◽  
The Given ◽  
Uniformly Continuous

In recent years, the Stone-Čech compactification of certain semigroups (e.g. discrete semigroups) has been an interesting semigroup compactification (i.e. a compact right semitopological semigroup which contains a dense continuous homomorphic image of the given semigroup) to study, because an Arens-type product can be introduced. If G is a non-compact and non-discrete locally compact abelian group, then it is not possible to introduce such a product into the Stone-Čech compactification βG of G (see [1]). However, let UC(G) be the Banach algebra of bounded uniformly continuous complex functions on G, and let UG be the spectrum of UC(G) with the Gelfand topology. If f∈ UC(G), then the functions f and fy defined on G byare also in UC(G).

Fatou–Zygmund sets

1973 ◽  
Vol 73 (1) ◽  
pp. 57-65
Author(s):  
Kenneth A. Ross
Keyword(s):  
Abelian Group ◽  
Trigonometric Polynomial ◽  
Compact Abelian Group ◽  
Measurable Subset ◽  
Algebraic Condition ◽  
Character Group ◽  
The Real ◽  
Sidon Sets ◽  
Image Position ◽  
Complex Valued

1. Let G be a compact Abelian group with character group X. Let be an increasing sequence of finite symmetric subsets of X, and consider a symmetric subset P of . For any Hermitian complex-valued function u on P, we write snu for the real-valued trigonometric polynomial . Edwards, Hewitt and Ross(4) investigated the following property for a non-void measurable subset W of G satisfying W ⊂ (int W)−:The validity of this implication was shown to be independent of the choice of . Accordingly, if (*) holds, P is called an FZ(W)-set. If P is an FZ(W)-set for all W, then P is termed a full FZ-set or full Fatou-Zygmund set. In this paper, we characterize the full FZ-sets as FZ(G)-sets satisfying a certain algebraic condition. In particular, we show that if G is connected, then a symmetric subset of X is an FZ(G)-set if and only if it is a full FZ-set. Some of the techniques are adaptations of those of Mme Déchamps-Gondim(1), (2). The class of full FZ-sets is not always closed under the operation of finite unions; this contrasts with the situation for Sidon sets and for FZ(G)-sets.

A diophantine problem on groups. III

1971 ◽  
Vol 70 (1) ◽  
pp. 31-47 ◽  
Author(s):  
R. C. Baker
Keyword(s):  
Abelian Group ◽  
Finite Order ◽  
Compact Abelian Group ◽  
Dual Group ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Diophantine Problem ◽  
Precise Sense ◽  
Image Position ◽  
Almost All

AbstractThe following generalization of a theorem of Weyl appeared in part I of this series of papers. Let G be a locally compact Abelian group with dual group ĝ. Let be a sequence in ĝ, not too slowly growing in a certain precise sense. Then, provided ĝ has ‘not too many’ elements of finite order, the sequencesare uniformly distributed on the circle, for almost all x in G.

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