Odometer actions on G-measures

1991 ◽  
Vol 11 (2) ◽  
pp. 279-307 ◽  
Author(s):  
Gavin Brown ◽  
Anthony H. Dooley
Keyword(s):  
Harmonic Analysis ◽  
Ergodic Theory ◽  
Finite Groups ◽  
Riesz Product ◽  
Direct Sum ◽  
New Methods ◽  
Product Construction ◽  
Ergodic Properties ◽  
Systematic Development

AbstractThe introduction of results from harmonic analysis leads to new methods in the study of the ergodic properties of measures under the action of the direct sum of finite groups. We take the first steps in a systematic development of part of ergodic theory based on the formalism of the Riesz product construction.

Representation Theory and Harmonic Analysis of Wreath Products of Finite Groups

2009 ◽  
Author(s):  
Tullio Ceccherini-Silberstein ◽  
Fabio Scarabotti ◽  
Filippo Tolli
Keyword(s):  
Representation Theory ◽  
Harmonic Analysis ◽  
Finite Groups ◽  
Wreath Products

Sylow Theory for a Certain Class of Operator Groups

1961 ◽  
Vol 13 ◽  
pp. 192-200 ◽  
Author(s):  
Christine W. Ayoub
Keyword(s):  
Direct Product ◽  
Finite Groups ◽  
Complete Lattice ◽  
Classical Group ◽  
Additive Group ◽  
Sylow Subgroups ◽  
Direct Sum

In this paper we consider again the group-theoretic configuration studied in (1) and (2). Let G be an additive group (not necessarily abelian), let M be a system of operators for G, and let ϕ be a family of admissible subgroups which form a complete lattice relative to intersection and compositum. Under these circumstances we call G an M — ϕ group. In (1) we studied the normal chains for an M — ϕ group and the relation between certain normal chains. In (2) we considered the possibility of representing an M — ϕ group as the direct sum of certain of its subgroups, and proved that with suitable restrictions on the M — ϕ group the analogue of the following theorem for finite groups holds: A group is the direct product of its Sylow subgroups if and only if it is nilpotent. Here we show that under suitable hypotheses (hypotheses (I), (II), and (III) stated at the beginning of §3) it is possible to generalize to M — ϕ groups many of the Sylow theorems of classical group theorem.

A reconstruction theorem for complex polynomials

2015 ◽  
Vol 26 (09) ◽  
pp. 1550073 ◽  
Author(s):  
Luka Boc Thaler
Keyword(s):  
Real Axis ◽  
Ergodic Theory ◽  
Complex Dynamics ◽  
Julia Set ◽  
Complex Polynomials ◽  
Analogous Statement ◽  
The Real ◽  
Ergodic Properties ◽  
Reconstruction Theorem ◽  
Exceptional Polynomials

Recently Takens' Reconstruction Theorem was studied in the complex analytic setting by Fornæss and Peters [Complex dynamics with focus on the real part, to appear in Ergodic Theory Dynam. Syst.]. They studied the real orbits of complex polynomials, and proved that for non-exceptional polynomials ergodic properties such as measure theoretic entropy are carried over to the real orbits mapping. Here we show that the result from [Complex dynamics with focus on the real part, to appear in Ergodic Theory Dynam. Syst.] also holds for exceptional polynomials, unless the Julia set is entirely contained in an invariant vertical line, in which case the entropy is 0. In [The reconstruction theorem for endomorphisms, Bull. Braz. Math. Soc. (N.S.) 33(2) (2002) 231–262.] Takens proved a reconstruction theorem for endomorphisms. In this case the reconstruction map is not necessarily an embedding, but the information of the reconstruction map is sufficient to recover the (2m + 1) th image of the original map. Our main result shows an analogous statement for the iteration of generic complex polynomials and the projection onto the real axis.

INFINITE-DIMENSIONAL GENERALIZED CONTINUED FRACTIONS, DISTRIBUTION OF QUADRATIC RESIDUES AND NON-RESIDUES, AND ERGODIC THEORY

2002 ◽  
Vol 05 (04) ◽  
pp. 555-570
Author(s):  
L. D. PUSTYL'NIKOV
Keyword(s):  
Prime Number ◽  
Ergodic Theory ◽  
Continued Fractions ◽  
Classical Problem ◽  
Infinite Dimensional ◽  
Ergodic Properties ◽  
Quadratic Residues ◽  
Generalized Continued Fractions

A new theory of generalized continued fractions for infinite-dimensional vectors with integer components is constructed. The results of this theory are applied to the classical problem on the distribution of quadratic residues and non-residues modulo a prime number and are based on the study of ergodic properties of some infinite-dimensional transformations.

Randall Dougherty and Alexander S. Kechris. The complexity of antidifferentiation. Advances in mathematics, vol. 88 (1991), pp. 145–169. - Ferenc Beleznay and Matthew Foreman. The collection of distal flows is not Borel. American journal of mathematics, vol. 117 (1995), pp. 203–239. - Ferenc Beleznay and Matthew Foreman. The complexity of the collection of measure-distal transformations. Ergodic theory and dynamical systems, vol. 16 (1996), pp. 929–962. - Howard Becker. Pointwise limits of subsequences and sets. Fundamenta mathematicae, vol. 128 (1987), pp. 159–170. - Howard Becker, Sylvain Kahane, and Alain Louveau. Some complete sets in harmonic analysis. Transactions of the American Mathematical Society, vol. 339 (1993), pp. 323–336. - Robert Kaufman. PCA sets and convexity Fundamenta mathematicae, vol. 163 (2000), pp. 267–275). - Howard Becker. Descriptive set theoretic phenomena in analysis and topology. Set theory of the continuum, edited by H. Judah, W. Just, and H. Woodin, Mathematical Sciences Research Institute publications, vol. 26, Springer-Verlag, New York, Berlin, Heidelberg, etc., 1992, pp. 1–25.

2001 ◽  
Vol 7 (3) ◽  
pp. 385-388
Author(s):  
Gabriel Debs
Keyword(s):  
New York ◽  
Dynamical Systems ◽  
Harmonic Analysis ◽  
Ergodic Theory ◽  
Set Theory ◽  
American Mathematical Society ◽  
And Topology ◽  
Mathematical Sciences ◽  
Descriptive Set ◽  
The Continuum

On the Decomposition of Groups

1969 ◽  
Vol 21 ◽  
pp. 762-768 ◽  
Author(s):  
Paul Hill
Keyword(s):  
Abelian Group ◽  
Direct Summand ◽  
Finite Groups ◽  
Commutative Case ◽  
Direct Sum ◽  
Infinite Subgroup

The problem in which we are interested is the following. Call an additively written group G finitely decomposable if G = Σ Gi is the weak sum of finite groups Gi, Consider the following property.Property P. Each subgroup of G having cardinality less than G is contained in a finitely decomposable direct summand of G.Does Property P imply that G is finitely decomposable? We shall demonstrate that the answer is negative even in the commutative case. Our question is closely related to (1, Problem 5). In (4), an abelian group is called a Fuchs 5-group if every infinite subgroup of the group can be embedded in a direct summand of the same cardinality. The question of whether or not a Fuchs 5-group is in fact a direct sum of countable groups has been open for several years.

scholarly journals Genomic Abelian Finite Groups

2021 ◽  
Author(s):  
Robersy Sanchez ◽  
Jesus Barreto
Keyword(s):  
Finite Groups ◽  
Genomic Medicine ◽  
Experimental Studies ◽  
Canonical Decomposition ◽  
Genome Architecture ◽  
Biological Processes ◽  
New Horizons ◽  
Direct Sum ◽  
Genomic Scale ◽  
Natural Domains

Experimental studies reveal that genome architecture splits into natural domains suggesting a well-structured genomic architecture, where, for each species, genome populations are integrated by individual mutational variants. Herein, we show that the architecture of population genomes from the same or closed related species can be quantitatively represented in terms of the direct sum of homocyclic abelian groups defined on the genetic code, where populations from the same species lead to the same canonical decomposition into p -groups.  This finding unveils a new ground for the application of the abelian group theory to genomics and epigenomics, opening new horizons for the study of the biological processes (at genomic scale) and provides new lens for genomic medicine.

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