Analytic cell decomposition and analytic motivic integration

The main results of this paper are a Cell Decomposition Theorem for Henselian valued fields with analytic structure in an analytic Denef-Pas language, and its application to analytic motivic integrals and analytic integrals over Fq((t)) of big enough characteristic. To accomplish this, we introduce a general framework for Henselian valued fields K with analytic structure, and we investigate the structure of analytic functions in one variable, defined on annuli over K. We also prove that, after parameterization, definable analytic functions are given by terms. The results in this paper pave the way for a theory of analytic motivic integration and analytic motivic constructible functions in the line of R. Cluckers and F. Loeser [Fonctions constructible et intégration motivique I, Comptes rendus de l'Académie des Sciences 339 (2004) 411-416].     

Weakly K-analytic spaces and the three-space property for analyticity

Let (E,E^′) be a dual pair of vector spaces. The paper studies general conditions which allow to lift analyticity (or K-analyticity) from the weak topology σ(E,E^′) to stronger ones in the frame of (E,E^′). First we show that the Mackey dual of a space Cp(X) is analytic iff the space X is countable. This yields that for uncountable analytic spaces X the Mackey dual of Cp(X) is weakly analytic but not analytic. We show that the Mackey dual E of an (LF)-space of a sequence of reflexive separable Fréchet spaces with the Heinrich density condition is analytic, i.e. E is a continuous image of the Polish space NN. This extends a result of Valdivia. We show also that weakly quasi-Suslin locally convex Baire spaces are metrizable and complete (this extends a result of De Wilde and Sunyach). We provide however a large class of weakly analytic but not analytic metrizable separable Baire topological vector spaces (not locally convex!). This will be used to prove that analyticity is not a three-space property but we show that a metrizable topological vector space E is analytic if E contains a complete locally convex analytic subspace F such that the quotient E/F is analytic. Several questions, remarks and examples are included.     

Existence of analytic solutions to analytic nonlinear q-difference equations

In this paper, some analytic approaches are formulated for the existence of analytic solutions of analytic nonlinear difference equations. From the point of view of dynamical systems, analytic solutions of such kinds of equations can be generally expressed by formal power series of exponential variables, so we are interested in considering a difference equation as a q-difference equation via a suitable coordinate transformation. After stating analytic results for formal series solutions of nonlinear q-difference equations, we also derive some results for the existence of analytic solutions to autonomous rational difference equations.     

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.     

Global analytic geometry

Motivated by the well-known lack of archimedean information in algebraic geometry, we define, formalizing Ostrowski's classification of seminorms on Z, a new type of valuation of a ring that combines the notion of Krull valuation with that of a multiplicative seminorm. This definition partially restores the broken symmetry between archimedean and non-archimedean valuations artificially introduced in arithmetic geometry by the theory of schemes. This also allows us to define a notion of global analytic space that reconciles Berkovich's notion of analytic space of a (Banach) ring with Huber's notion of non-archimedean analytic spaces. After defining natural generalized valuation spectra and computing the spectrum of Z and Z[X], we define analytic spectra and sheaves of analytic functions on them.     

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.     

Maximal independent, analytic sets in Abelian polish groups

The paper deals with the questions:

1. whether a topological module admits maximal linearly independent subsets that are analytic
2. whether an Abelian topological group admits maximal independent subsets that are analytic
3. whether a topological field extension admits transcendence bases that are analytic.

     

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.

     

Deformations of trianalytic subvarieties of hyperkähler manifolds

Let M be a compact complex manifold equipped with a hyperk?hler metric, and X be a closed complex analytic subvariety of M. In alg-geom 9403006, we proved that X is trianalytic (i.e., complex analytic with respect to all complex structures induced by the hyperk?hler structure), provided that M is generic in its deformation class. Here we study the complex analytic deformations of trianalytic subvarieties. We prove that all deformations of X are trianalytic and naturally isomorphic to X as complex analytic varieties. We show that this isomorphism is compatible with the metric induced from M. Also, we prove that the Douady space of complex analytic deformations of X in M is equipped with a natural hyperk?hler structure.     

Wave packet analysis of Schrödinger equations in analytic function spaces

We consider a class of linear Schrödinger equations in R^d${\mathbb{R}}^d$, with analytic symbols. We prove a global-in-time integral representation for the corresponding propagator as a generalized Gabor multiplier with a window analytic and decaying exponentially at infinity, which is transported by the Hamiltonian flow. We then provide three applications of the above result: the exponential sparsity in phase space of the corresponding propagator with respect to Gabor wave packets, a wave packet characterization of Fourier integral operators with analytic phases and symbols, and the propagation of analytic singularities.     

Hyperbolicity, transversality and analytic first integrals

Let A be a (normally) hyperbolic compact invariant manifold of an analytic diffeomorphism f of an analytic manifold M. We assume that the stable and unstable manifold of A intersect transversally (in an admissible way), the dynamics on A is ergodic and the modulus of the eigenvalues associated to the stable and unstable manifold, respectively, satisfy a non-resonance condition. In the case where A is a point or a torus, we prove that the discrete dynamical system associated to f does not admit an analytic first integral. The proof is based on a triviality lemma, which is of combinatorial nature, and a geometrical lemma. The same techniques, allow us to prove analytic non-integrability of Hamiltonian systems having Arnold diffusion. In particular, using results of Xia, we prove analytic non-integrability of the elliptic restricted three-body problem, as well as the planar three-body problem.     

Robot navigation functions on manifolds with boundary

This paper concerns the construction of a class of scalar valued analytic maps on analytic manifolds with boundary. These maps, which we term navigation functions, are constructed on an arbitrary sphere world—a compact connected subset of Euclidean n-space whose boundary is formed from the disjoint union of a finite number of (n − l)-spheres. We show that this class is invariant under composition with analytic diffeomorphisms: our sphere world construction immediately generates a navigation function on all manifolds into which a sphere world is deformable. On the other hand, certain well known results of S. Smale guarantee the existence of smooth navigation functions on any smooth manifold. This suggests that analytic navigation functions exist, as well, on more general analytic manifolds than the deformed sphere worlds we presently consider.     

An analytic version of the De Rham theorem

LetV be a real analytic paracompact variety; in §1 of this paper we prove that: $$H^q (V,R) \approx \frac{{closed analytic differentiable q - forms on V}}{{exact analytic differentiable q - forms on V}}$$ Then we prove that the closed (exact) analytic differentiableq-forms onV are dense, in the Whitney topology, in the set of closed (exact) differentiableq-forms onV. We also consider the problem of extending closed (exact) analytic differentiableq-forms, defined on a subvarietyX ofV, to closed (exact) analytic forms defined onV.     

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.     

The reversibility and the center problem

In this work we study the narrow relation between reversibility and the center problem and also between reversibility and the integrability problem. It is well known that an analytic system having either a non-degenerate or nilpotent center at the origin is analytically reversible or orbitally analytically reversible, respectively. In this paper we prove the existence of a smooth map that transforms an analytic system having a degenerate center at the origin with either an analytic first integral or a C^∞ inverse integrating factor into a reversible linear system (after rescaling the time). Moreover, if the degenerate center has an analytic or a C^∞ reversing symmetry, then the transformed system by the map also has a reversing symmetry. From the knowledge of a first integral near the center we give a procedure to detect reversing symmetries.     

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.     

Homogeneous spaces and transitive actions by analytic groups

If X is homogeneous, analytic, and strongly locally homogeneous,then there is an analytic group acting transitively on X. Thereis an example of an analytic space on which some separable metrizablegroup acts transitively, but on which no analytic group actstransitively.     

Applications of branched values to p-adic functional equations on analytic functions

Let \(\mathbb{K}\) be an algebraically closed field of characteristic 0, complete with respect to an ultrametric absolute value. Results on branched values obtained in a previous paper are used to prove that algebraic functional equations of the form g ^q = hf ^q + w have no solution among transcendental entire functions f, g or among unbounded analytic functions inside an open disk, when w is a polynomial or a bounded analytic function and h is a polynomial or an analytic function whose zeros are of order multiple of q. We also show that an analytic function whose zeros are multiple of an integer q inside a disk is the q-th power of another analytic function, provided q is a prime to the residue characteristic.