Birkhoff's Theorem Research Articles

Periodic solutions for a dumbbell satellite equation

In this paper, we study the existence of at least two geometrically distinct periodic solutions for a differential equation which models the planar oscillations of a dumbbell satellite under the influence of the gravity field generated by an oblate body, considering the effect of the zonal harmonic parameter $$J_{2}$$. And at least one of such two periodic solutions is unstable. The proof is based on the version of the Poincare–Birkhoff theorem due to Franks. Moreover, we also study the existence and multiplicity of periodic solutions and subharmonic solutions with winding number.

A Poincaré–Birkhoff theorem for Hamiltonian flows on nonconvex domains

We present a higher-dimensional version of the Poincaré–Birkhoff theorem which applies to Poincaré time maps of Hamiltonian systems. The maps under consideration are neither required to be close to the identity nor to have a monotone twist. The annulus is replaced by the product of an N-dimensional torus and the interior of a (N−1)-dimensional (not necessarily convex) embedded sphere; on the other hand, the classical boundary twist condition is replaced by an avoiding rays condition.

Periodic solutions of second order equations via rotation numbers

We consider the problem of the existence and multiplicity of periodic solutions associated to a class of scalar equations of the form x″+f(t,x)=0. The class considered is such that the behaviour of its solutions near zero and infinity can be compared two suitable piecewise linear systems. We show how a rotation number approach, together with the Poincaré–Birkhoff theorem and the phase-plane analysis of the spiral properties, allows to obtain multiplicity results in terms of the gap between the rotation numbers of the referred piecewise linear systems at zero and at infinity. These systems may also be resonant. In particular, our approach can be used to deal with the problems without both global existence of the Cauchy problems associated to the equation and the sign assumption on f. The typical example is a partially superlinear second order equation. Our main result generalizes some classical results of Jacobowitz and Hartman, among others.

The basic ergodic theorems, yet again

A generalization of Rokhlin’s Tower Lemma is presented. The Maximal Ergodic Theorem is then obtained as a corollary. We also use the generalized Rokhlin lemma, this time combined with a subadditive version of Kac’s formula, to deduce a subadditive version of the Maximal Ergodic Theorem due to Silva and Thieullen. In both the additive and subadditive cases, these maximal theorems immediately imply that “heavy” points have positive probability. We use heaviness to prove the pointwise ergodic theorems of Birkhoff and Kingman.

Can the universe be described by a wave function?

Suppose we assume that in gently curved spacetime (a) causality is not violated to leading order (b) the Birkhoff theorem holds to leading order and (c) CPT invariance holds. Then we argue that the “mostly empty” universe we observe around us cannot be described by an exact wave function [Formula: see text]. Rather, the weakly coupled particles we see are approximate quasiparticles arising as excitations of a “fuzz”. The “fuzz” does have an exact wave function [Formula: see text], but this exact wave function does not directly describe local particles. The argument proceeds by relating the cosmological setting to the black hole information paradox, and then using the small corrections theorem to show the impossibility of an exact wave function describing the visible universe.

Birkhoff’s theorem in f(R) theory of gravity

Birkhoff’s theorem in general relativity states that every spherically symmetric solution of Einstein field equations in vacuum is either static or Schwarzschild. This theorem has been established in the scalar-tensor theories of gravity by Reddy (J. Phys. A 6, 1867 (1973)). In this paper, we prove this theorem in f(R) theory of gravity (where R Ricci scalar) when the scalaron $\Phi$ (R) of the theory is time-independent only. However, no attention is given to the order of the f(R) theory. Here it is important to note that the Birkhoff theorem in f(R) gravity states only that if R is time-independent, the spherical solution is static (not Schwarzschild).

Pointwise convergence of Birkhoff averages for global observables.

It is well-known that a strict analogue of the Birkhoff Ergodic Theorem in infinite ergodic theory is trivial; it states that for any infinite-measure-preserving ergodic system, the Birkhoff average of every integrable function is almost everywhere zero. Nor does a different rescaling of the Birkhoff sum that leads to a non-degenerate pointwise limit exist. In this paper, we give a version of Birkhoff's theorem for conservative, ergodic, infinite-measure-preserving dynamical systems where instead of integrable functions we use certain elements of , which we generically call global observables. Our main theorem applies to general systems but requires a hypothesis of "approximate partial averaging" on the observables. The idea behind the result, however, applies to more general situations, as we show with an example. Finally, by means of counterexamples and numerical simulations, we discuss the question of finding the optimal class of observables for which a Birkhoff theorem holds for infinite-measure-preserving systems.

Vacua of exotic massive 3D gravity

We consider the recently proposed exotic 3D massive gravity. We show that this theory has a rich space of vacua, including asymptotically Anti de-Sitter (AdS) geometries obeying either the standard Brown-Henneaux boundary conditions or the weakened asymptotic behavior of the so-called Log-gravity. Both sectors contain non-Einstein spaces with SO(2) × mathbb{R} isometry group, showing that the Birkhoff theorem does not hold all over the parameter space, even if strong AdS boundary conditions are imposed. Some of these geometries correspond to 3D black holes dressed with a Log-gravity graviton. We conjecture that such geometries appear in a curve of the parameter space where the exotic 3D massive gravity on AdS3 is dual to a chiral conformal field theory. The theory also contains other interesting vacua, including different families of non-AdS black holes.

Revisiting the Birkhoff theorem from a dual null point of view

The Birkhoff theorem is a well-known result in general relativity and it is used in many applications. However, its most general version, due to Bona, is almost unknown and presented in a form less accessible to the relativist and cosmologist community. Moreover, many wield it mistakenly as a simple transposition of Newton's iron sphere theorem. In the present work, we propose a modern, dual null, presentation --- useful in many explorations, including black holes --- of the theorem that renders accessible most of the results of Bona's version. In addition, we discuss the fluid contents admissible for the application of the theorem, beyond a vacuum, and we demonstrate how the formalism greatly simplifies solving the dynamical equations and allows one to express the solution as a power expansion in $r$. We present a family of solutions that share the properties predicted by the Birkhoff theorem and discuss the existence of trapped and antitrapped regions. The formalism manifestly shows how the type of region --- trapped or untrapped --- determines the character of the Killing vector.

Unbounded Convergence in the Convergence Vector Lattices: a Survey

Various convergences in vector lattices were historically a subject of deep investigation which stems from the begining of the 20th century in works of Riesz, Kantorovich, Nakano, Vulikh, Zanen, and many other mathematicians. The study of the unbounded order convergence had been initiated by Nakano in late 40th in connection with Birkhoff's ergodic theorem. The idea of Nakano was to define the almost everywhere convergence in terms of lattice operations without the direct use of measure theory. Many years later it was recognised that the unbounded order convergence is also rathe useful in probability theory. Since then, the idea of investigating of convergences by using their unbounded versions, have been exploited in several papers. For instance, unbounded convergences in vector lattices have attracted attention of many researchers in order to find new approaches to various problems of functional analysis, operator theory, variational calculus, theory of risk measures in mathematical finance, stochastic processes, etc. Some of those unbounded convergences, like unbounded norm convergence, unbounded multi-norm convergence, unbounded $\tau$-convergence are topological. Others are not topological in general, for example: the unbounded order convergence, the unbounded relative uniform convergence, various unbounded convergences in lattice-normed lattices, etc. Topological convergences are, as usual, more flexible for an investigation due to the compactness arguments, etc. The non-topological convergences are more complicated in genelal, as it can be seen on an example of the a.e-convergence. In the present paper we present recent developments in convergence vector lattices with emphasis on related unbounded convergences. Special attention is paid to the case of convergence in lattice multi pseudo normed vector lattices that generalizes most of cases which were discussed in the literature in the last 5 years.

Routine Psychological Testing of the Individual Is Not Valid.

In this article, we present and argue our assertion that current routine psychological testing of individuals is not valid. To support our assertion, we review the concept of ergodicity, Birkhoff's theorem, and Molenaar's manifesto, which together support our contention that the direct transposition of population estimations for producing inferences about the individual is not valid. We argue that this practice of direct transposition is the root cause of why routine psychological testing of individual is not valid. We then provide an example of a common application of psychological testing of an individual, explaining why this practice is not valid. Finally, we discuss how the intraindividual (or within-person) approach provides some prospect for valid individual testing and also introduces new challenges. We hope that our questioning of current psychological testing practices motivates researchers to propose and study novel methodological propositions to address the issues raised by our assertion.

Regularized Kalb-Ramond magnetic monopole with finite energy

In a previous work we suggested a self-gravitating electromagnetic monopole solution in a string-inspired model involving global spontaneous breaking of a SO(3) internal symmetry and Kalb-Ramond (KR) axions, stemming from an antisymmetric tensor field in the massless string multiplet. These axions carry a charge, which, in our model, also plays the role of the magnetic charge. The resulting geometry is close to that of a Reissner-Nordstr\"om (RN) black hole with charge proportional to the KR-axion charge. We proposed the existence of a thin shell structure surrounding a (large inner) core as the dominant mass contribution to the energy functional. Although the resulting energy was finite, and proportional to the KR-axion charge, the size of the shell was not determined and left as a phenomenological parameter. In the current article, we propose a new way to calculate the size of the thin shell: string theory considerations suggest that the short-distance physics inside the inner core may be dominated by a positive cosmological constant term proportional to the scale of the spontaneous symmetry breaking of SO(3). The size of the shell is estimated by matching the RN metric of the shell to the de Sitter metric inside the core. The matching entails the Israel junction conditions for the metric and its first derivatives at the inner boundary of the shell, and determines the inner mass-shell radius. The axion charge plays an important role in guaranteeing the positivity of the ``mass coefficient'' of the gravitational potential term appearing in the metric; so, the KR electromagnetic monopole shows normal attractive gravitational effects. This is to be contrasted with the axion-less global monopole case where such a matching is known to yield a negative ``mass coefficient'' (and, hence, a repulsive gravitational effect). The total energy of our monopole within the shell is calculated. As a result of the violation of Birkhoff's theorem (due to the formal divergence of the energy functional in the absence of a large distance cutoff), the total energy does not have to equal the mass coefficient. However, for phenomenologically relevant sets of parameters, the ratio of the total energy and the mass coefficient in the shell is close to 1. The gravitational ``effective mass coefficient'' in the shell can be made equal to the total energy outside the core by a small decrease in the cosmological constant in the de Sitter region. This is achieved through a dilaton potential which is suitably negative inside the de-Sitter region, but vanishes outside that region.

Periodic solutions with prescribed minimal period of vortex type problems in domains

We consider Hamiltonian systems with two degrees of freedom of point vortex type for in a domain . In the classical point vortex context the Hamiltonian is of the form where is the regular part of a hydrodynamic Green function in Ω, is the Robin function: , and , are the vortex strengths. We prove the existence of infinitely many periodic solutions with prescribed minimal period that are superpositions of a slow motion of the center of vorticity close to a star-shaped level line of h and of a fast rotation of the two vortices around their center of vorticity. The proofs are based on a recent higher dimensional version of the Poincaré–Birkhoff theorem due to Fonda and Ureña.

Violation of Birkhoff's theorem for pure quadratic gravity action

We investigate the application of Birkhoff's theorem to Einstein's quadratic conjecture for the vacuum action. A derivation of the vacuum solution follows from the general metric ansatz for an uncharged, spherically symmetric and static stellar object. It is shown that the Schwarzschild‐de Sitter metric is not the only static solution for the quadratic field equation in vacuum. Moreover, with a time dependency as starting point in the metric ansatz, we find a time‐dependent solution. The existence of such a solution implies that Birkhoff's theorem is not strictly valid for the quadratic gravity Lagrangian.

A uniform Birkhoff theorem

Garret Birkhoff's HSP theorem characterizes the classes of models of algebraic theories as those being closed with respect to homomorphic images, subalgebras, and products. In particular, it implies that an algebra $\mathbf{B}$ satisfies all equations that hold in an algebra $\mathbf{A}$ of the same type if and only if $\mathbf{B}$ is a homomorphic image of a subalgebra of a (possibly infinite) direct power of $\mathbf{A}$. The former statement is equivalent to the existence of a natural map sending term functions of the algebra $\mathbf{A}$ to those of $\mathbf{B}$, and it is natural to wonder about continuity properties of this mapping. We show that this map is uniformly continuous if and only if every finitely generated subalgebra of $\mathbf{B}$ is a homomorphic image of a subalgebra of a finite power of $\mathbf{A}$ -- without any additional assumptions concerning the algebras $\mathbf{A}$ and $\mathbf{B}$. Moreover, provided that $\mathbf{A}$ is almost locally finite (for instance if $\mathbf{A}$ is locally oligomorphic or locally finite), the considered map is uniformly continuous if and only if it is Cauchy-continuous. In particular, our results extend a recent theorem by Bodirsky and Pinsker beyond the countable $\omega$-categorical setting.

Quantifying the Role of Folding in Nonautonomous Flows: The Unsteady Double-Gyre

We analyze chaos in the well-known nonautonomous Double-Gyre system. A key focus is on folding, which is possibly the less-studied aspect of the “stretching+folding=chaos” mantra of chaotic dynamics. Despite the Double-Gyre not having the classical homoclinic structure for the usage of the Smale–Birkhoff theorem to establish chaos, we use the concept of folding to prove the existence of an embedded horseshoe map. We also show how curvature of manifolds can be used to identify fold points in the Double-Gyre. This method is applicable to general nonautonomous flows in two dimensions, defined for either finite or infinite times.

Subharmonic solutions of Hamiltonian systems displaying some kind of sublinear growth

AbstractWe prove the existence and multiplicity of subharmonic solutions for Hamiltonian systems obtained as perturbations ofNplanar uncoupled systems which, e.g., model some type of asymmetric oscillators. The nonlinearities are assumed to satisfy Landesman–Lazer conditions at the zero eigenvalue, and to have some kind of sublinear behavior at infinity. The proof is carried out by the use of a generalized version of the Poincaré–Birkhoff Theorem. Different situations, including Lotka–Volterra systems, or systems with singularities, are also illustrated.

Ergodic Theory, Dynamic Mode Decomposition, and Computation of Spectral Properties of the Koopman Operator

We establish the convergence of a class of numerical algorithms, known as dynamic mode decomposition (DMD), for computation of the eigenvalues and eigenfunctions of the infinite-dimensional Koopman operator. The algorithms act on data coming from observables on a state space, arranged in Hankel-type matrices. The proofs utilize the assumption that the underlying dynamical system is ergodic. This includes the classical measure-preserving systems, as well as systems whose attractors support a physical measure. Our approach relies on the observation that vector projections in DMD can be used to approximate the function projections by the virtue of Birkhoff's ergodic theorem. Using this fact, we show that applying DMD to Hankel data matrices in the limit of infinite-time observations yields the true Koopman eigenfunctions and eigenvalues. We also show that the singular value decomposition, which is the central part of most DMD algorithms, converges to the proper orthogonal decomposition of observables. We use this result to obtain a representation of the dynamics of systems with continuous spectrum based on the lifting of the coordinates to the space of observables. The numerical application of these methods is demonstrated using well-known dynamical systems and examples from computational fluid dynamics.

АЛГЕБРАИЧЕСКИЙ ТИП СФЕРИЧЕСКИ СИММЕТРИЧНОГО ГРАВИТАЦИОННОГО ПОЛЯ И ТЕОРЕМА БИРКГОФА

АЛГЕБРАИЧЕСКИЙ ТИП СФЕРИЧЕСКИ СИММЕТРИЧНОГО ГРАВИТАЦИОННОГО ПОЛЯ И ТЕОРЕМА БИРКГОФА

The Birkhoff theorem for unitary matrices of arbitrary dimensions

It was shown recently that Birkhoff's theorem for doubly stochastic matrices can be extended to unitary matrices with equal line sums whenever the dimension of the matrices is prime. We prove a generalization of the Birkhoff theorem for unitary matrices with equal line sums for arbitrary dimension.