Poisson-stable point

The central sub-class of non-wandering points are points which are stable in the sense of Poisson. The main feature of a Poisson-stable point is not only the recurrence of its neighborhood but the recurrence of the trajectory itself. The definition of Poisson-stable points below is different in some ways but equivalent to the definition given in Chap. 1. 

Definition 7.5. A point xq is said to be positively stable in the sense of Poisson -stable) if there exists a sequence where tn +oo asn -hoo, such that... 

The definition of a negative Poisson stable (P -stable) point is analogous to the above except that tn - oo here. In the case where the point Xq is both -stable and P -stable, it is said to be stable in the sense of Poisson. 

In the preceding sections, we have discussed the set of center motions. In essence, we have found that it is the closure of the set of Poisson-stable trajectories. It does not exclude the case where the latter ones may simply be periodic orbits. But if there is a single Poisson-stable unclosed trajectory, then by virtue of Birkhoff s theorem in Sec. 1.2, there is a continuum of Poisson-stable trajectories. As for the rest of the trajectories in the center, it is known that the set of points which are not Poisson-stable is the union of not more... 

The Poisson-stable trajectories may be sub-divided into two kinds depending on whether the sequence Tfc(e) of Poincare return times of a P-trajectory to its -neighborhood is bounded or not. Birkhoff named the trajectories of the first kind recurrent trajectories. Such a trajectory is remarkable because regardless of the choice of the initial point, given e > 0 the whole trajectory lies in an -neighborhood of the segment of the trajectory corresponding to a time interval L(e). Obviously, equilibrium states and periodic orbits are the closed recurrent trajectories. 

The closure of an unclosed Poisson-stable trajectory whose return times are unbounded for some e > 0 is called a quasiminimal set. A quasiminimal set contains, besides Poisson-stable trajectories which are dense everywhere in it, some other invariant and closed subsets. These may be equilibrium states, periodic orbits, non-resonant invariant tori, other minimal sets, homoclinic and heteroclinic orbits, etc., among which a P-trajectory is wandering. This gives a clue to why the recurrent times of the non-trivial unclosed P-trajectory are unbounded. Furthermore, this also points out that Poisson-stable trajectories of a quasiminimal set, due to their unpredictable behavior in time, are of... 

Soft white, ductile metal high-purity metal is very ductile at ordinary temperatures occurs in three allotropic forms (i) body-centered cubic form, alpha iron stable up to 910°C, (ii) face-centered cubic form, gamma iron occurring between 910 to 1,390°C, and (iii) body-centered delta iron allotrope forming above 1,390°C. Density 7.873 g/cm at 20°C melting point 1,538°C vaporizes at 2,861°C hardness (Brinell) 60 electrical resistivity 4.71 microhm-cm at 0°C tensile strength 30,000 psi Poisson s ratio 0.29 modulus of elasticity 28.5 X 10 psi thermal neutron absorption cross-section 2.62 bams velocity of sound 5,130 m/s at 20°C. 

Fig. 2. Plot of the relative proportionality constant a as a function of the saturation (65,536 counts). Each data point represents at least 10 photoelectrons thus the uncertainty due to Poisson statistics is 10 . The data rvere accumulated using a very stable incandescent lamp and various accurately gated integration times. Plot (a) is the result of 42 tests of 5 different diodes across the array. In each test, the number of counts for a given exposure was compared with the number of counts recorded for a reference exposure corresponding to approximately 50% of saturation. The a value drops suddenly beyond 96% saturation. Plot (6) depicts the same data as (a) on an expanded scale in order to show the rapid dropoff of a as the percent of satination exceeds 96%. (Reprinted with permission from Menningen etal., 1995a, Contrib. Plasma Phys. 35, 359, 1995 Wiley-VCH, Inc.)...

An issue with the PCM formalism introduced in Section 11.2.2.1 is that the electrostatic energy is in general a discontinuous function as the solute atoms are displaced, because the number and size of the surface tesserae may change as a function of solute geometry. A similar problem is suffered by finite-difference Poisson-Boltzmann solvers, and the "solution" in those cases (in order to achieve stable forces for MD simulations, for example) is tight thresholding and/or some kind of interpolation between grid points [80-83]. 

Since each point on a P-trajectory is non-wandering, this result is also valid for points stable in the sense of Poisson. The closing lenuna implies the following meaningful corollary a rough system with a P-trajectory possesses infinitely many periodic orbits. 

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