An algebraic characterization of holomorphic nondegeneracy for real algebraic hypersurfaces and its application to CR mappings

We give a new algebraic characterization of holomorphic nondegeneracy for embedded real algebraic hypersurfaces in , . We then use this criterion to prove the following result about real analyticity of smooth CR mappings: any smooth CR mapping H between a real analytic hypersurface and a rigid polynomial holomorphically nondegenerate hypersurface is real analytic, provided the map H is not totally degenerate in the sense of Baouendi and Rothschild. Received September 19, 1997     

Closure and connected component of a planar global semianalytic set defined by analytic functions definable in o-minimal structure

We consider a global semianalytic set defined by real analytic functions definable in an o-minimal structure. When the o-minimal structure is polynomially bounded, we show that the closure of this set is a global semianalytic set defined by definable real analytic functions. We also demonstrate that a connected component of a planar global semianalytic set defined by real analytic functions definable in a substructure of the restricted analytic field is a global semianalytic set defined by definable real analytic functions.     

Regularity of a free boundary problem

Let u be the Newtonian potential of a real analytic distribution in an open set Ω. In this paper we assume u is analytically continuable from the complement of Ω into some neighborhood of a point x₀ ∈ ∂Ω, and we study conditions under which the analytic continuation implies that ∂Ω is a real analytic hypersurface in some neighborhood of x₀.     

某类实解析与复解析分歧问题的R-有限决定及K-有限决定问题的注记

该文给出了某类实解析与复解析分歧问题是不能R-有限决定的．并利用上述结果证明了一类分岐问题是不能K-有限决定的．同时给出了一个K-有限决定但不能R-有限决定的个例子,它说明在分岐问题中R-有限决定与K-有限决定不是互为充要的条件．     

On the linear representations of circle expanding maps

In this paper, it is proved that the two real analytic expanding endomorphisms of the unit circle are equivalent if and only if they have equal eigenvalues along corresponding cycles. A sufficient and necessary condition that a real analytic solution of Cvitanovic-Feigenbaum equation induces a real analytic expanding map is given. Supported by the NSFC.     

The principal axis theorem for holomorphic functions

An algebraic approach to Rellich's theorem is given which states that any analytic family of matrices which is normal on the real axis can be diagonalized by an analytic family of matrices which is unitary on the real axis. We show that this result is a special version of a purely algebraic theorem on the diagonalization of matrices over fields with henselian valuations.

     

Extension of Fréchet valued real analytic functions from subvarieties of ℝd

It is shown that for an algebraic subvariety X of ℝ^d every Fréchet valued real analytic function on X can be extended to a real analytic function on ℝ^d if and only if X is of type (PL), i.e. all of its singularities are of a certain type. Necessity of this condition is shown for any real analytic variety. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Globally analytic hypoelliptic vector fields on compact surfaces

We present a characterization of the global analytic hypoellipticity of a complex, non-singular, real analytic vector field defined on a compact, connected, orientable, two-dimensional, real analytic manifold.

In particular, we show that such vector fields exist only on the torus.

     

Proof of the Hyperplane Zeros Conjecture of Lagarias and Wang

We prove that a real analytic subset of a torus group that is contained in its image under an expanding endomorphism is a finite union of translates of closed subgroups. This confirms the hyperplane zeros conjecture of Lagarias and Wang for real analytic varieties. Our proof uses real analytic geometry, topological dynamics, and Fourier analysis.      

Real Analytic Fibre Bundles: Some Results on Classification and Homotopies

The classification problem for holomorphic fibre bundles over Stein spaces was solved by H. GRAUERT. Along the same lines, the real coherent analytic case was considered by A. TOGNOLI and V. ANCONA. In this paper we propose a different approach, based on classifying spaces, in order to study the previous problem for real analytic fibre bundles over C -analytic subspaces of R ^m. So, let X be a C -analytic subspace of R ^m and G a compact Lie group. The main result is a characterization of the real analytic G-principal fibre bundles over X for which the analytic and topological equivalence coincide. Moreover, we prove that these bundles can be classified also by means of homotopy classes of analytic maps of X into classifying spaces. Among the others results, are worth recording: a relative approximation theorem of continuous cross sections by analytic ones, a theorem about the equivalence between analytical and topological homotopy between cross sections and a covering homotopy theorem.     

A class of real cocycles having an analytic coboundary modification

We define here a certain class of procedures (a.a.c.c.p.) for constructing real valued cocycles over irrational rotations. Each such procedure is realizable over a residual set of possible rotations, and we prove that each such cocycle is cohomologous to a real analytic cocycle. The procedure in Section 3 of [10] is seen to be of this type and hence not only is cohomologous toC ^∞ as is shown there, but is actually cohomologous to a real analytic cocycle. We also show that following the method of [6] a procedure can be given to obtain rank-1 Anzai skew products of mixed spectral type that are real analytic. Research supported by KBN grant 512/2/91. Research supported by KBN grant 512/2/91. Research supported by NSF grant DMS 01524351.     

Blowing-up orientato

Summary In this paper we define a new kind of blowing-up, as a functor from the category of real analytic spaces to the category of real semianalytic spaces, in such a way that orientability is preserved. Then we prove an existence theorem for ?oriented blowing-ups? of real analytic spaces.

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Riassunto In questo articolo si definisce un nuovo tipo blowing-up, come un funtore dalla categoria degli spazi analitici reali alla categoria degli spazi semianalitici reali, in modo tale che l'orientabilità sia preservata. Si dimostra quindi un teorema di esistenza per blowing-ups orientati di spazi analitici reali.

     

Regular or stochastic dynamics in real analytic families of unimodal maps

In this paper we prove that in any non-trivial real analytic family of quasiquadratic maps, almost any map is either regular (i.e., it has an attracting cycle) or stochastic (i.e., it has an absolutely continuous invariant measure). To this end we show that the space of analytic maps is foliated by codimension-one analytic submanifolds, hybrid classes. This allows us to transfer the regular or stochastic property of the quadratic family to any non-trivial real analytic family. To Jacob Palis on his 60th birthdayMathematics Subject Classification (2000) 37E05, 37F45, 30D05     

Ergodic transport theory and piecewise analytic subactions for analytic dynamics

We consider a piecewise analytic real expanding map f: [0, 1] ?? [0, 1] of degree d which preserves orientation, and a real analytic positive potential g: [0, 1] ?? ?. We assume the map and the potential have a complex analytic extension to a neighborhood of the interval in the complex plane. We also assume log g is well defined for this extension. It is known in Complex Dynamics that under the above hypothesis, for the given potential ?? log g, where ?? is a real constant, there exists a real analytic eigenfunction ? _?? defined on [0, 1] (with a complex analytic extension) for the Ruelle operator of ?? log g. Under some assumptions we show that $\frac{1} {\beta }\log \varphi _\beta$ converges and is a piecewise analytic calibrated subaction. Our theory can be applied when log g(x) = ?log f??(x). In that case we relate the involution kernel to the so called scaling function.     

KAM-tori near an analytic elliptic fixed point

We study the accumulation of an elliptic fixed point of a real analytic Hamiltonian by quasi-periodic invariant tori. We show that a fixed point with Diophantine frequency vector ω ₀ is always accumulated by invariant complex analytic KAM-tori. Indeed, the following alternative holds: If the Birkhoff normal form of the Hamiltonian at the invariant point satisfies a Rüssmann transversality condition, the fixed point is accumulated by real analytic KAM-tori which cover positive Lebesgue measure in the phase space (in this part it suffices to assume that ω ₀ has rationally independent coordinates). If the Birkhoff normal form is degenerate, there exists an analytic subvariety of complex dimension at least d + 1 passing through 0 that is foliated by complex analytic KAM-tori with frequency ω ₀. This is an extension of previous results obtained in [1] to the case of an elliptic fixed point.     

Characterization of the generic unfolding of a weak focus

In this paper we prove that the orbital class of a generic real analytic family unfolding a weak focus is determined by the conjugacy class of its Poincaré monodromy and vice versa. We solve the embedding problem by means of quasiconformal surgery on the formal normal form. The surgery yields an integrable abstract almost complex 2-manifold equipped with an elliptic foliation. The monodromy of the latter coincides with the second iterate of a germ of prescribed family of real analytic diffeomorphisms undergoing a flip bifurcation.     

A Transfer Principle for the Continuations of Real Functions to the Levi-Civita Field

We discuss the properties of the continuations of real functions to the Levi-Civita field. In particular, we show that, whenever a function f is analytic on a compact interval [a, b] ? ?, f and its analytic continuation f?_∞ satisfy the same properties that can be expressed in the language of real closed ordered fields. If f is not analytic, then this equivalence does not hold. These results suggest an analogy with the internal and external functions of nonstandard analysis: while the canonical continuations of analytic functions resemble internal functions, the continuations of non-analytic functions behave like external functions. Inspired by this analogy, we suggest some directions for further research.     

Global Analytic Regularity for Structures of Co-Rank One

We consider real analytic involutive structures 𝒱, of co-rank one, defined on a real analytic paracompact orientable manifold M. To each such structure we associate certain connected subsets of M which we call the level sets of 𝒱. We prove that analytic regularity propagates along them. With a further assumption on the level sets of 𝒱 we characterize the global analytic hypoellipticity of a differential operator naturally associated to 𝒱.

As an application we study a case of tube structures.     

The parabolic analytic functions and the derivative of real functions

Among the bidimensional hypercomplex-number systems defined as the parabolic (dual) numbers are introduced with the rule α = 0. As well as the functions of a complex variable, the analytic functions of a parabolic variable can be introduced as analytic continuation of the real functions of a real variable. These functions hold the property that the “imaginary” part is linked to the derivative of the “real” part. In this paper we will show how this property allows one to demonstrate in an algebraic way some rules of the differential calculus for the real functions of a real variable.     

Integrability of Analytic Almost Complex Structures on Banach Manifolds

We prove that the classical integrability condition for almost complex structures on finite-dimensional smooth manifolds also works in infinite dimensions in the case of almost complex structures that are real analytic on real analytic Banach manifolds. With this result at hand, we extend some known results concerning existence of invariant complex structures on homogeneous spaces of Banach–Lie groups. By way of illustration, we construct the complex flag manifolds associated with unital C^*-algebras.Mathematics Subject Classifications (2000): primary 32Q60; secondary 53C15, 58B12.