Search results for "Mathematics - Complex Variables"
showing 10 items of 53 documents
Optimal Extensions of Conformal Mappings from the Unit Disk to Cardioid-Type Domains
2019
AbstractThe conformal mapping $$f(z)=(z+1)^2 $$ f ( z ) = ( z + 1 ) 2 from $${\mathbb {D}}$$ D onto the standard cardioid has a homeomorphic extension of finite distortion to entire $${\mathbb {R}}^2 .$$ R 2 . We study the optimal regularity of such extensions, in terms of the integrability degree of the distortion and of the derivatives, and these for the inverse. We generalize all outcomes to the case of conformal mappings from $${\mathbb {D}}$$ D onto cardioid-type domains.
Weyl symbols and boundedness of Toeplitz operators
2019
International audience; We study Toeplitz operators on the Bargmann space, with Toeplitz symbols that are exponentials of inhomogeneous quadratic polynomials. It is shown that the boundedness of such operators is implied by the boundedness of the corresponding Weyl symbols.
Controlled diffeomorphic extension of homeomorphisms
2018
Let $\Omega$ be an internal chord-arc Jordan domain and $\varphi:\mathbb S\rightarrow\partial\Omega$ be a homeomorphism. We show that $\varphi$ has finite dyadic energy if and only if $\varphi$ has a diffeomorphic extension $h: \mathbb D\rightarrow \Omega$ which has finite energy.
Mapping properties for the Bargmann transform on modulation spaces
2010
We investigate mapping properties for the Bargmann transform and prove that this transform is isometric and bijective from modulation spaces to convenient Banach spaces of analytic functions.
Redundant Picard–Fuchs System for Abelian Integrals
2001
We derive an explicit system of Picard-Fuchs differential equations satisfied by Abelian integrals of monomial forms and majorize its coefficients. A peculiar feature of this construction is that the system admitting such explicit majorants, appears only in dimension approximately two times greater than the standard Picard-Fuchs system. The result is used to obtain a partial solution to the tangential Hilbert 16th problem. We establish upper bounds for the number of zeros of arbitrary Abelian integrals on a positive distance from the critical locus. Under the additional assumption that the critical values of the Hamiltonian are distant from each other (after a proper normalization), we were…
Blenders near polynomial product maps of $\mathbb C^2$
2021
In this paper we show that if $p$ is a polynomial which bifurcates then the product map $(z,w)\mapsto(p(z),q(w))$ can be approximated by polynomial skew products possessing special dynamical objets called blenders. Moreover, these objets can be chosen to be of two types : repelling or saddle. As a consequence, such product map belongs to the closure of the interior of two different sets : the bifurcation locus of $H_d(\mathbb P^2)$ and the set of endomorphisms having an attracting set of non-empty interior. In an independent part, we use perturbations of H\'enon maps to obtain examples of attracting sets with repelling points and also of quasi-attractors which are not attracting sets.
Analysis of complex singularities in high-Reynolds-number Navier-Stokes solutions
2013
AbstractNumerical solutions of the laminar Prandtl boundary-layer and Navier–Stokes equations are considered for the case of the two-dimensional uniform flow past an impulsively-started circular cylinder. The various viscous–inviscid interactions that occur during the unsteady separation process are investigated by applying complex singularity analysis to the wall shear and streamwise velocity component of the two solutions. This is carried out using two different methodologies, namely a singularity-tracking method and the Padé approximation. It is shown how the van Dommelen and Shen singularity that occurs in solutions of the Prandtl boundary-layer equations evolves in the complex plane be…
Envelopes of open sets and extending holomorphic functions on dual Banach spaces
2010
We investigate certain envelopes of open sets in dual Banach spaces which are related to extending holomorphic functions. We give a variety of examples of absolutely convex sets showing that the extension is in many cases not possible. We also establish connections to the study of iterated weak* sequential closures of convex sets in the dual of separable spaces.
Mean ergodic composition operators on Banach spaces of holomorphic functions
2016
[EN] Given a symbol cc, i.e., a holomorphic endomorphism of the unit disc, we consider the composition operator C-phi(f) = f circle phi defined on the Banach spaces of holomorphic functions A(D) and H-infinity(D). We obtain different conditions on the symbol phi which characterize when the composition operator is mean ergodic and uniformly mean ergodic in the corresponding spaces. These conditions are related to the asymptotic behavior of the iterates of the symbol. Finally, we deal with some particular case in the setting of weighted Banach spaces of holomorphic functions.
Equilibrium measures for uniformly quasiregular dynamics
2012
We establish the existence and fundamental properties of the equilibrium measure in uniformly quasiregular dynamics. We show that a uniformly quasiregular endomorphism $f$ of degree at least 2 on a closed Riemannian manifold admits an equilibrium measure $\mu_f$, which is balanced and invariant under $f$ and non-atomic, and whose support agrees with the Julia set of $f$. Furthermore we show that $f$ is strongly mixing with respect to the measure $\mu_f$. We also characterize the measure $\mu_f$ using an approximation property by iterated pullbacks of points under $f$ up to a set of exceptional initial points of Hausdorff dimension at most $n-1$. These dynamical mixing and approximation resu…