scholarly journals Homeomorphisms of S1 and Factorization

2019 ◽  
Author(s):  
Mark Dalthorp ◽  
Doug Pickrell
Keyword(s):  
Fourier Series ◽  
Finite Type ◽  
Parameter Family ◽  
Complex Parameter ◽  
Triangular Factorization ◽  
Linear Fractional Transformations ◽  
Root Subgroup ◽  
Decay Properties ◽  
Group Of Homeomorphisms ◽  
Verblunsky Coefficients

Abstract For each $n>0$ there is a one complex parameter family of homeomorphisms of the circle consisting of linear fractional transformations “conjugated by $z \to z^n$”. We show that these families are free of relations, which determines the structure of “the group of homeomorphisms of finite type”. We next consider factorization for more robust groups of homeomorphisms. We refer to this as root subgroup factorization (because the factors correspond to root subgroups). We are especially interested in how root subgroup factorization is related to triangular factorization (i.e., conformal welding) and correspondences between smoothness properties of the homeomorphisms and decay properties of the root subgroup parameters. This leads to interesting comparisons with Fourier series and the theory of Verblunsky coefficients.

A case of distinction between Fourier integrals and Fourier series

1927 ◽  
Vol 23 (7) ◽  
pp. 755-767
Author(s):  
Margaret Eleanor Grimshaw
Keyword(s):  
Fourier Series ◽  
Finite Type ◽  
Generating Function ◽  
Fourier Integral ◽  
Finite Range ◽  
Fourier Integrals

A Fourier integral is said to be of finite type if its generating function vanishes for all sufficiently large values of ¦x¦. Because the coefficient functions are defined by integrals over a finite range, the behaviour of such a Fourier integral usually resembles closely that of the corresponding series.

On a Transformation Group

1969 ◽  
Vol 21 ◽  
pp. 935-941 ◽  
Author(s):  
S. K. Kaul
Keyword(s):  
Finite Order ◽  
Metric Space ◽  
Transformation Group ◽  
Real Numbers ◽  
Compact Metric Space ◽  
Topological Transformation ◽  
Linear Fractional Transformations ◽  
Group Of Homeomorphisms

0. Let Γ denote a group of real linear fractional transformations (the constants defining any element of Γ are real numbers); see (3, § 2, p. 10). Then it is known that Γ is discontinuous if and only if it is discrete (3, Theorem 2F, p. 13).Now Γ may also be regarded, equivalently, as a group of homeomorphisms of a disc D onto itself; and if Γ is discrete, then, except for elements of finite order, each element of Γ is either of type 1 or type 2 (see Definitions 0.1 and 0.2 below).We wish to generalize the result quoted above in purely topological terms. Thus, throughout this paper we denote by X a compact metric space with metric d, and by G a topological transformation group on X each element of which, except the identity e, is either of type 1 or type 2. Let L = ﹛a ∈ X: g(a) = a for some g in G — e﹜, and . We assume furthermore that 0 is non-empty.

scholarly journals Mating quadratic maps with the modular group II

2019 ◽  
Vol 220 (1) ◽  
pp. 185-210
Author(s):  
Shaun Bullett ◽  
Luna Lomonaco
Keyword(s):  
Modular Group ◽  
Parameter Family ◽  
Quadratic Polynomial ◽  
Complex Parameter ◽  
Rational Maps ◽  
Quadratic Maps ◽  
The Family ◽  
The One ◽  
Holomorphic Correspondences ◽  
Connectedness Locus

Abstract In 1994 S. Bullett and C. Penrose introduced the one complex parameter family of (2 : 2) holomorphic correspondences $$\mathcal {F}_a$$Fa: $$\begin{aligned} \left( \frac{aw-1}{w-1}\right) ^2+\left( \frac{aw-1}{w-1}\right) \left( \frac{az+1}{z+1}\right) +\left( \frac{az+1}{z+1}\right) ^2=3 \end{aligned}$$aw-1w-12+aw-1w-1az+1z+1+az+1z+12=3and proved that for every value of $$a \in [4,7] \subset \mathbb {R}$$a∈[4,7]⊂R the correspondence $$\mathcal {F}_a$$Fa is a mating between a quadratic polynomial $$Q_c(z)=z^2+c,\,\,c \in \mathbb {R}$$Qc(z)=z2+c,c∈R, and the modular group $$\varGamma =PSL(2,\mathbb {Z})$$Γ=PSL(2,Z). They conjectured that this is the case for every member of the family $$\mathcal {F}_a$$Fa which has a in the connectedness locus. We show here that matings between the modular group and rational maps in the parabolic quadratic family $$Per_1(1)$$Per1(1) provide a better model: we prove that every member of the family $$\mathcal {F}_a$$Fa which has a in the connectedness locus is such a mating.

Loops in SL(2, ℂ) and root subgroup factorization

2018 ◽  
Vol 07 (03) ◽  
pp. 1850008
Author(s):  
Estelle Basor ◽  
Doug Pickrell
Keyword(s):  
Critical Exponent ◽  
Toeplitz Operators ◽  
Rational Functions ◽  
Triangular Factorization ◽  
Exceptional Set ◽  
Determinant Formula ◽  
Root Subgroup ◽  
Toeplitz Determinant ◽  
Simple Generalization ◽  
Broad Outline

In previous work, we proved that for a [Formula: see text]-valued loop having the critical degree of smoothness (one half of a derivative in the [Formula: see text] Sobolev sense), the following are equivalent: (1) the Toeplitz and shifted Toeplitz operators associated to the loop are invertible, (2) the loop has a unique triangular factorization, and (3) the loop has a unique root subgroup factorization. For a loop [Formula: see text] satisfying these conditions, the Toeplitz determinant [Formula: see text] and shifted Toeplitz determinant [Formula: see text] factor as products in root subgroup coordinates. In this paper, we observe that, at least in broad outline, there is a relatively simple generalization to loops having values in [Formula: see text]. The main novel features are that (1) root subgroup coordinates are now rational functions, i.e. there is an exceptional set and associated uniqueness issues, and (2) the noncompactness of [Formula: see text] entails that loops are no longer automatically bounded, and this (together with the exceptional set) complicates the analysis at the critical exponent.

EUCLIDEANIZATION OF MAJORANA AND WEYL FERMIONS

1991 ◽  
Vol 06 (30) ◽  
pp. 2811-2817 ◽  
Author(s):  
MAYANIK R. MEHTA
Keyword(s):  
Minkowski Space ◽  
Degrees Of Freedom ◽  
Parameter Family ◽  
Complex Parameter ◽  
Space Theory ◽  
Yang Mills ◽  
Euclidean Action ◽  
Weyl Fermions

A continuous procedure is presented for euclideanization of Majorana and Weyl fermions without doubling their degrees of freedom. The Euclidean theory so obtained is SO (4) invariant and Osterwalder-Schrader (OS) positive. This enables us to define a one-complex parameter family of the N=1 supersymmetric Yang-Mills (SSYM) theories which interpolate between the Minkowski and a Euclidean SSYM theory. The interpolating action, and hence the Euclidean action, manifests all the continous symmetries of the original Minkowski space theory.

Constructions of Non-Commutative Lp-Spaces with a Complex Parameter Arising from Modular Actions

1997 ◽  
Vol 08 (08) ◽  
pp. 1029-1066 ◽  
Author(s):  
Hideaki Izumi
Keyword(s):  
Banach Spaces ◽  
Von Neumann Algebra ◽  
Interpolation Method ◽  
Parameter Family ◽  
Complex Parameter ◽  
Complex Interpolation ◽  
Von Neumann ◽  
Lp Spaces ◽  
Natural Maps ◽  
Complete Extension

For a von Neumann algebra ℳ and a weight φ on ℳ, we will construct a complex one-parameter family [Formula: see text] of non-commutative Lp-spaces by using Calderón's complex interpolation method. This is a simultaneous and complete extension of the construction of non-commutative Lp-spaces by H. Kosaki and M. Terp. Moreover, we will show that for each p, all the parametrized Lp-spaces are mutually isometrically isomorphic as Banach spaces via natural maps.

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