Quasisymmetric map

Last updated

In mathematics, a quasisymmetric homeomorphism between metric spaces is a map that generalizes bi-Lipschitz maps. While bi-Lipschitz maps shrink or expand the diameter of a set by no more than a multiplicative factor, quasisymmetric maps satisfy the weaker geometric property that they preserve the relative sizes of sets: if two sets A and B have diameters t and are no more than distance t apart, then the ratio of their sizes changes by no more than a multiplicative constant. These maps are also related to quasiconformal maps, since in many circumstances they are in fact equivalent. [1]

Contents

Definition

Let (X, dX) and (Y, dY) be two metric spaces. A homeomorphism f:X  Y is said to be η-quasisymmetric if there is an increasing function η : [0, ∞)  [0, ∞) such that for any triple x, y, z of distinct points in X, we have

Basic properties

Inverses are quasisymmetric
If f : X  Y is an invertible η-quasisymmetric map as above, then its inverse map is -quasisymmetric, where
Quasisymmetric maps preserve relative sizes of sets
If and are subsets of and is a subset of , then

Examples

Weakly quasisymmetric maps

A map f:X→Y is said to be H-weakly-quasisymmetric for some if for all triples of distinct points in , then

Not all weakly quasisymmetric maps are quasisymmetric. However, if is connected and and are doubling, then all weakly quasisymmetric maps are quasisymmetric. The appeal of this result is that proving weak-quasisymmetry is much easier than proving quasisymmetry directly, and in many natural settings the two notions are equivalent.

δ-monotone maps

A monotone map f:H  H on a Hilbert space H is δ-monotone if for all x and y in H,

To grasp what this condition means geometrically, suppose f(0) = 0 and consider the above estimate when y = 0. Then it implies that the angle between the vector x and its image f(x) stays between 0 and arccos δ < π/2.

These maps are quasisymmetric, although they are a much narrower subclass of quasisymmetric maps. For example, while a general quasisymmetric map in the complex plane could map the real line to a set of Hausdorff dimension strictly greater than one, a δ-monotone will always map the real line to a rotated graph of a Lipschitz function L:ℝ  ℝ. [2]

Doubling measures

The real line

Quasisymmetric homeomorphisms of the real line to itself can be characterized in terms of their derivatives. [3] An increasing homeomorphism f:ℝ  ℝ is quasisymmetric if and only if there is a constant C > 0 and a doubling measure μ on the real line such that

Euclidean space

An analogous result holds in Euclidean space. Suppose C = 0 and we rewrite the above equation for f as

Writing it this way, we can attempt to define a map using this same integral, but instead integrate (what is now a vector valued integrand) over ℝn: if μ is a doubling measure on ℝn and

then the map

is quasisymmetric (in fact, it is δ-monotone for some δ depending on the measure μ). [4]

Quasisymmetry and quasiconformality in Euclidean space

Let and be open subsets of ℝn. If f : Ω  Ω´ is η-quasisymmetric, then it is also K-quasiconformal, where is a constant depending on .

Conversely, if f : Ω  Ω´ is K-quasiconformal and is contained in , then is η-quasisymmetric on , where depends only on .

Quasi-Möbius maps

A related but weaker condition is the notion of quasi-Möbius maps where instead of the ratio only the cross-ratio is considered: [5]

Definition

Let (X, dX) and (Y, dY) be two metric spaces and let η : [0, ∞)  [0, ∞) be an increasing function. An η-quasi-Möbius homeomorphism f:X  Y is a homeomorphism for which for every quadruple x, y, z, t of distinct points in X, we have

See also

Related Research Articles

<span class="mw-page-title-main">Harmonic function</span> Functions in mathematics

In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function where U is an open subset of that satisfies Laplace's equation, that is,

<span class="mw-page-title-main">Lorentz group</span> Lie group of Lorentz transformations

In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicist Hendrik Lorentz.

In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered in the 1750s by Swiss mathematician Leonhard Euler and Italian mathematician Joseph-Louis Lagrange.

<span class="mw-page-title-main">Jensen's inequality</span> Theorem of convex functions

In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. It was proved by Jensen in 1906, building on an earlier proof of the same inequality for doubly-differentiable functions by Otto Hölder in 1889. Given its generality, the inequality appears in many forms depending on the context, some of which are presented below. In its simplest form the inequality states that the convex transformation of a mean is less than or equal to the mean applied after convex transformation; it is a simple corollary that the opposite is true of concave transformations.

In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the algebra produces the Hodge dual of the element. This map was introduced by W. V. D. Hodge.

In mathematics, the Cauchy principal value, named after Augustin-Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined. In this method, a singularity on an integral interval is avoided by limiting the integral interval to the non singular domain.

In physics, the Polyakov action is an action of the two-dimensional conformal field theory describing the worldsheet of a string in string theory. It was introduced by Stanley Deser and Bruno Zumino and independently by L. Brink, P. Di Vecchia and P. S. Howe in 1976, and has become associated with Alexander Polyakov after he made use of it in quantizing the string in 1981. The action reads:

In differential geometry, the four-gradient is the four-vector analogue of the gradient from vector calculus.

In mathematical complex analysis, a quasiconformal mapping, introduced by Grötzsch (1928) and named by Ahlfors (1935), is a homeomorphism between plane domains which to first order takes small circles to small ellipses of bounded eccentricity.

In physics, the Majorana equation is a relativistic wave equation. It is named after the Italian physicist Ettore Majorana, who proposed it in 1937 as a means of describing fermions that are their own antiparticle. Particles corresponding to this equation are termed Majorana particles, although that term now has a more expansive meaning, referring to any fermionic particle that is its own anti-particle.

In mathematics, a π-system on a set is a collection of certain subsets of such that

In mathematics, the Poincaré inequality is a result in the theory of Sobolev spaces, named after the French mathematician Henri Poincaré. The inequality allows one to obtain bounds on a function using bounds on its derivatives and the geometry of its domain of definition. Such bounds are of great importance in the modern, direct methods of the calculus of variations. A very closely related result is Friedrichs' inequality.

In mathematics and mathematical physics, raising and lowering indices are operations on tensors which change their type. Raising and lowering indices are a form of index manipulation in tensor expressions.

<span class="mw-page-title-main">Peridynamics</span>

Peridynamics is a non-local formulation of continuum mechanics that is oriented toward deformations with discontinuities, especially fractures. Originally, bond-based peridynamic has been introduced, wherein, internal interaction forces between a material point and all the other ones with which it can interact, are modeled as a central forces field. This type of force fields can be imagined as a mesh of bonds connecting each point of the body with every other interacting point within a certain distance which depends on material property, called peridynamic horizon. Later, to overcome bond-based framework limitations for the material Poisson’s ratio, state-base peridynamics, has been formulated. Its characteristic feature is that the force exchanged between a point and another one is influenced by the deformation state of all other bonds relative to its interaction zone.

In mathematics, the spectral theory of ordinary differential equations is the part of spectral theory concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation. In his dissertation, Hermann Weyl generalized the classical Sturm–Liouville theory on a finite closed interval to second order differential operators with singularities at the endpoints of the interval, possibly semi-infinite or infinite. Unlike the classical case, the spectrum may no longer consist of just a countable set of eigenvalues, but may also contain a continuous part. In this case the eigenfunction expansion involves an integral over the continuous part with respect to a spectral measure, given by the Titchmarsh–Kodaira formula. The theory was put in its final simplified form for singular differential equations of even degree by Kodaira and others, using von Neumann's spectral theorem. It has had important applications in quantum mechanics, operator theory and harmonic analysis on semisimple Lie groups.

In mathematics, the Beltrami equation, named after Eugenio Beltrami, is the partial differential equation

In mathematics, a quasicircle is a Jordan curve in the complex plane that is the image of a circle under a quasiconformal mapping of the plane onto itself. Originally introduced independently by Pfluger (1961) and Tienari (1962), in the older literature they were referred to as quasiconformal curves, a terminology which also applied to arcs. In complex analysis and geometric function theory, quasicircles play a fundamental role in the description of the universal Teichmüller space, through quasisymmetric homeomorphisms of the circle. Quasicircles also play an important role in complex dynamical systems.

In mathematics, in the field of complex analysis, a Nevanlinna function is a complex function which is an analytic function on the open upper half-plane and has non-negative imaginary part. A Nevanlinna function maps the upper half-plane to itself or to a real constant, but is not necessarily injective or surjective. Functions with this property are sometimes also known as Herglotz, Pick or R functions.

In quantum computing, Mølmer–Sørensen gate scheme refers to an implementation procedure for various multi-qubit quantum logic gates used mostly in trapped ion quantum computing. This procedure is based on the original proposition by Klaus Mølmer and Anders Sørensen in 1999-2000.

In mathematics, Rathjen's  psi function is an ordinal collapsing function developed by Michael Rathjen. It collapses weakly Mahlo cardinals to generate large countable ordinals. A weakly Mahlo cardinal is a cardinal such that the set of regular cardinals below is closed under . Rathjen uses this to diagonalise over the weakly inaccessible hierarchy.

References

  1. Heinonen, Juha (2001). Lectures on Analysis on Metric Spaces. Universitext. New York: Springer-Verlag. pp. x+140. ISBN   978-0-387-95104-1.
  2. Kovalev, Leonid V. (2007). "Quasiconformal geometry of monotone mappings". Journal of the London Mathematical Society. 75 (2): 391–408. CiteSeerX   10.1.1.194.2458 . doi:10.1112/jlms/jdm008.
  3. Beurling, A.; Ahlfors, L. (1956). "The boundary correspondence under quasiconformal mappings". Acta Math. 96: 125–142. doi: 10.1007/bf02392360 .
  4. Kovalev, Leonid; Maldonado, Diego; Wu, Jang-Mei (2007). "Doubling measures, monotonicity, and quasiconformality". Math. Z. 257 (3): 525–545. arXiv: math/0611110 . doi:10.1007/s00209-007-0132-5. S2CID   119716883.
  5. Buyalo, Sergei; Schroeder, Viktor (2007). Elements of Asymptotic Geometry. EMS Monographs in Mathematics. American Mathematical Society. p. 209. ISBN   978-3-03719-036-4.