A generalization of Poincaré's theorem for recurrence systems

A new proof and a genuine generalization to systems of first order equations is given from Poincaré classical theorem on ratio asymptotics of solutions of higher order recurrence equations. The asymptotic behavior of a fundamental system of solutions is obtained.     

Poincaré's recurrence theorem and the unitarity of the S-matrix

A scattering process can be described by suitably closing the system and considering the first return map from the entrance onto itself. This scattering map may be singular and discontinuous, but it will be measured preserving as a consequence of the recurrence theorem applied to any region of a simpler map. In the case of a billiard this is the Birkhoff map. The semiclassical quantization of the Birkhoff map can be subdivided into an entrance and a repeller. The construction of a scattering operator then follows in exact analogy to the classical process. Generically, the approximate unitarity of the semiclassical Birkhoff map is inherited by the S-matrix, even for highly resonant scattering where direct quantization of the scattering map breaks down.     

Parametric Poincaré-Perron theorem with applications

We prove a parametric generalization of the classical Poincaré-Perron theorem on stabilizing recurrence relations, where we assume that the varying coefficients of a recurrence depend on auxiliary parameters and converge uniformly in these parameters to their limiting values. As an application, we study convergence of the ratios of families of functions satisfying finite recurrence relations with varying functional coefficients. For example, we explicitly describe the asymptotic ratio for two classes of biorthogonal polynomials introduced by Ismail and Masson.     

Poincaré's Lemma on Some Non-Euclidean Structures

The author proves the Poincar′e lemma on some(n + 1)-dimensional corank1 sub-Riemannian structures, formulating the(n-1)n(n~2+3 -2)/8 necessarily and sufficient y "curl-vanishing" compatibility conditions. In particular, this result solves partially an open problem formulated by Calin and Chang. The proof in this paper is based on a Poincar′e lemma stated on Riemannian manifolds and a suitable Ces`aro-Volterra path integral formula established in local coordinates. As a byproduct, a Saint-Venant lemma is also provided on generic Riemannian manifolds. Some examples are presented on the hyperbolic space and Carnot/Heisenberg groups.     

An alternative proof of Painlevé's theorem

In this article we show some aspects of analytical and numerical solution of the n-body problem, which arises from the classical Newtonian model for gravitation attraction. We prove the non-existence of stationary solutions and give an alternative proof for Painlevé's theorem.     

A topological version of the Poincaré-Birkhoff theorem with two fixed points

The main result of this paper gives a topological property satisfied by any homeomorphism of the annulus \mathbb A = \mathbb S¹ ×[-1, 1]{\mathbb {A} = \mathbb {S}^1 \times [-1, 1]} isotopic to the identity and with at most one fixed point. This generalizes the classical Poincaré-Birkhoff theorem because this property certainly does not hold for an area preserving homeomorphism h of \mathbb A{\mathbb {A}} with the usual boundary twist condition. We also have two corollaries of this result. The first one shows in particular that the boundary twist assumption may be weakened by demanding that the homeomorphism h has a lift H to the strip [(\mathbbA)\tilde] = \mathbbR ×[-1, 1]{\tilde{\mathbb{A}} = \mathbb{R} \times [-1, 1]} possessing both a forward orbit unbounded on the right and a forward orbit unbounded on the left. As a second corollary we get a new proof of a version of the Conley–Zehnder theorem in \mathbb A{\mathbb {A}} : if a homeomorphism of \mathbb A{\mathbb {A}} isotopic to the identity preserves the area and has mean rotation zero, then it possesses two fixed points.     

Convergence acceleration of continued fractions of Poincaré's type 1

A new method of convergence acceleration is proposed for continued fractions of Poincaré's type 1. Each step of the method (and not only the first one, as in the Hautot method [1]) is based on an asymptotic behaviour of continued fraction tails. A theorem is proved detailing properties of the method in six cases considered here. Results of numerical tests for all Hautot's examples confirm a good performance of the method.     

A generalization of Gauss-Kuzmin-Lévy theorem

We prove a generalized Gauss-Kuzmin-Lévy theorem for the generalized Gauss transformation

$T_p\left(x\right)=\left\{\frac{p}{x}\right\}.$

In addition, we give an estimate for the constant that appears in the theorem.     

Strong Poincaré recurrence theorem in MV-algebras

The classical Poincaré strong recurrence theorem states that for any probability space (Ω, ℒ, P), any P-measure preserving transformation T, and any A ∈ ℒ, almost all points of A return to A infinitely many times. In the present paper the Poincaré theorem is proved when the σ-algebra ℒ is substituted by an MV-algebra of a special type. Another approach is used in [RIEČAN, B.: Poincaré recurrence theorem in MV-algebras. In: Proc. IFSA-EUSFLAT 2009 (To appear)], where the weak variant of the theorem is proved, of course, for arbitrary MV-algebras. Such generalizations were already done in the literature, e.g. for quantum logic, see [DVUREČENSKIJ, A.: On some properties of transformations of a logic, Math. Slovaca 26 (1976), 131–137.     

Nielsen theory, Floer homology and a generalisation of the Poincaré-Birkhoff theorem

The purpose of this mostly expository paper is to discuss a connection between Nielsen fixed point theory and symplectic Floer homology for symplectomorphisms of surfaces and a calculation of Seidel’s symplectic Floer homology for different mapping classes. We also describe symplectic zeta functions and an asymptotic symplectic invariant. A generalisation of the Poincaré-Birkhoff fixed point theorem and Arnold conjecture is proposed. Dedicated to Vladimir Igorevich Arnold