Physical Measures for Multivalued Inverse Iterates Near Hyperbolic Repellors

We study new invariant probability measures, describing the distribution of multivalued inverse iterates (i.e. of different local inverse iterates) for a non-invertible smooth function f which is hyperbolic, but not necessarily expanding on a repellor Λ. The methods for the higher dimensional non-expanding and non-invertible case are different than the ones for diffeomorphisms, due to the lack of a nice unstable foliation (local unstable manifolds depend on prehistories and may intersect each other, both in Λ and outside Λ), and the fact that Markov partitions may not exist on Λ. We obtain that for Lebesgue almost all points z in a neighbourhood V of Λ, the normalized averages of Dirac measures on the consecutive preimage sets of z converge weakly to an equilibrium measure μ ⁻ on Λ; this implies that μ ⁻ is a physical measure for the local inverse iterates of f. It turns out that μ ⁻ is an inverse SRB measure in the sense that it is the only invariant measure satisfying a Pesin type formula for the negative Lyapunov exponents. Also we show that μ ⁻ has absolutely continuous conditional measures on local stable manifolds, by using the above convergence of measures. We prove then that f:(Λ,ℬ(Λ),μ ⁻)→(Λ,ℬ(Λ),μ ⁻) cannot be one-sided Bernoulli, although it is an exact endomorphism of Lebesgue spaces. Several classes of examples of hyperbolic non-invertible and non-expanding repellors, with their inverse SRB measures, are given in the end.     

Crisis and unstable dimension variability in the bailout embedding map

The dynamics of inertial particles in 2-d incompressible flows can be modeled by 4-d bailout embedding maps. The density of the inertial particles, relative to the density of the fluid, is a crucial parameter which controls the dynamical behaviour of the particles. We study here the dynamical behaviour of aerosols, i.e. particles heavier than the flow. An attractor widening and merging crisis is seen in the phase space in the aerosol case. Crisis-induced intermittency is seen in the time series and the laminar length distribution of times before bursts give rise to a power law with the exponent β = −1/3. The maximum Lyapunov exponent near the crisis fluctuates around zero indicating unstable dimension variability (UDV) in the system. The presence of unstable dimension variability is confirmed by the behaviour of the probability distributions of the finite time Lyapunov exponents.      

Vacuum Instability

NAbstract Following fresh attempts to resolve the problem of the energy density of the vacuum, we reconsider the case where the cosmological constant is derived from a higher-dimensional version of general relativity, and interpret the gauge-dependence of Λ as a dynamical effect. This leads to a relation between the change in Λ and the line element (action) which is independent of gauge choices and fundamental constants: dΛds² = −6. This implies that the (classical) vacuum is unstable, with implications for particle production.     

Scales in nuclear matter: Chiral dynamics with pion nucleon form factors<Superscript>⋆</Superscript>

A systematic calculation of nuclear matter is performed which includes the long-range correlations between nucleons arising from one- and two-pion exchange. Three-body effects from 2π exchange with excitations of virtual Δ(1232)-isobars are also taken into account in our diagrammatic calculation of the energy per particle ˉ(k _f). In order to eliminate possible high-momentum components from the interactions we introduce at each pion-baryon vertex a form factor of monopole type. The empirical nuclear matter saturation point, ρ₀ ≃ 0.16fm^-3, ˉ₀ ≃ - 16MeV, is well reproduced with a monopole mass of Λ ≃ 4πf _π ≃ 1.16GeV. As in the recent approach based on the universal low-momentum NN potential V _low-k, the inclusion of three-body effects is crucial in order to achieve saturation of nuclear matter. We demonstrate that the dependence of the pion exchange contributions to ˉ(k _f) on the “resolution” scale Λ can be compensated over a wide range of Λ by counterterms with two “running” contact couplings. As a further application we study the in-medium chiral condensate 〈ˉq〉(ρ) beyond the linear density approximation. For ρ ⩽ 1.5ρ₀ we find small corrections from the derivative dˉ(k _f)/dm _π, which are stable against variations of the monopole regulator mass Λ.     

Study of FCNC mediated Z boson effectin the semileptonic rare baryonic decays Λ b → Λl + l -

We study the effect of the FCNC mediated Z boson in the rare semileptonic baryonic decays Λ_b → Λl⁺l^-. We consider the model where the standard model fermion sector is extended by an extra vector-like down quark, as a consequence of which it allows for CP-violating Z mediated flavor changing neutral current at the tree level. We find that due to this non-universal Zbs coupling, the branching ratios of the rare semileptonic Λ_b decays are enhanced reasonably from their corresponding standard model values and the zero point of the forward-backward asymmetry for Λ_b → Λ_μ⁺_μ^- is shifted to the left. Received: 2 June 2005, Published online: 26 October 2005 PACS: 13.30.Ce, 12.60.-i, 11.30.Hv     

Rare decays of Λ b → Λ γ and Λ b → Λ l + l ? in universal extra-dimension model

The exclusive weak decay of Λ _b →Λ γ and Λ _b →Λ l ⁺ l ⁻ are investigated in the Appelquist–Cheng–Dobrescu model, which is an extension of the standard model in the presence of universal extra dimensions. Employing the transition form factors obtained in the light-cone sum rules, we analyze how the invariant-mass distribution, the forward–backward asymmetry and the polarization asymmetry of the Λ baryon of these decay modes can be used to constrain the only additional free parameter with respect to the standard model, namely, the radius, R, of the extra dimension. Our results indicate that the Kaluza–Klein modes can lead to approximately 25% suppression of the branching ratio of Λ _b →Λ γ; however, their contributions can bring about 10% enhancement to the decay rate of Λ _b →Λ l ⁺ l ⁻. It is shown that in the present scenario the zero position of the forward–backward asymmetry of Λ _b →Λ μ ⁺ μ ⁻ is sensitive to the compactification parameter R, while the measurement of polarizations of Λ baryon in the Λ _b decays are not a useful tool to provide any valuable information for the universal extra-dimension model.     

Ergodic Theory of Parabolic Horseshoes

In this paper we develop the ergodic theory for a horseshoe map f which is uniformly hyperbolic, except at one parabolic fixed point ω and possibly also on W ^s (ω). We call f a parabolic horseshoe map. In order to analyze dynamical and geometric properties of such horseshoes, by making use of induced maps, we establish, in the context of σ-finite measures, an appropriate version of the variational principle for continuous potentials with mild distortion defined on subshifts of finite type. Staying in this setting, we propose a concept of σ-finite equilibrium states (each classical probability equilibrium state is a σ-finite equilibrium state). We then study the unstable pressure function , the corresponding finite and σ-finite equilibrium states and their associated conditional measures. The main idea is to relate the pressure function to the pressure of an embedded parabolic iterated function system and to apply the developed theory of the symbolic σ-finite thermodynamic formalism. We prove, in particular, an appropriate form of the Bowen-Ruelle-Manning-McCluskey formula, the existence of exactly two σ-finite ergodic conservative equilibrium states for the potential –t ^u log |Df|E ^u | (where t ^u denotes the unstable dimension), one of which is the Dirac δ-measure supported at the parabolic fixed point and the other being non-atomic. We also show that the conditional measures of this non-atomic equilibrium state on unstable manifolds, are equivalent to (finite and positive) packing measures, whereas the Hausdorff measures vanish. As an application of our results we obtain a classification for the existence of a generalized physical measure, as well as a criteria implying the non-existence of an ergodic measure of maximal dimension. The research of the first author was supported in part by the NSF Grant DMS 0400481. The research of the second author was supported in part by the National Science Foundation under Grant No. EPS-0236913 and matching support from the State of Kansas through Kansas Technology Enterprise Corporation.     

Stable Ergodic Properties of Cocycles¶over Hyperbolic Attractors

We consider a hyperbolic flow φ ^t defined on an attracting basic set Λ. A map from the first (Čech) cohomology group of Λ into the dynamic cohomology group is constructed. This map is used to discuss the stable ergodicity and mixing of compact Lie group extensions and velocity changes of φ ^t . Received: 17 June 1998 / Accepted: 24 February 1999     

Special electronic structures of inverse spinels LiMVO<Subscript>4</Subscript>(M = Ni and Cu): a first-principles study

We use the Ulam method to study spectral properties of the Perron-Frobenius operators of dynamical maps in a chaotic regime. For maps with absorption we show numerically that the spectrum is characterized by the fractal Weyl law recently established for nonunitary operators describing poles of quantum chaotic scattering with the Weyl exponent ν = d-1, where d is the fractal dimension of corresponding strange set of trajectories nonescaping in future times. In contrast, for dissipative maps we numerically find the Weyl exponent ν = d/2 where d is the fractal dimension of strange attractor. The Weyl exponent can be also expressed via the relation ν = d₀/2 where d₀ is the fractal dimension of the invariant sets. We also discuss the properties of eigenvalues and eigenvectors of such operators characterized by the fractal Weyl law.     

Light-cone QCD sum rules for the Λ baryon electromagnetic form factors and its magnetic moment

We present the light-cone QCD sum rules up to twist 6 for the electromagnetic form factors of the Λ baryon. To estimate the magnetic moment of the baryon, the magnetic form factor is fitted by the dipole formula. The numerical value of our estimation is μ _Λ =−(0.64±0.04)μ _N , which is in accordance with the experimental data and the existing theoretical results. We find that it is twist 4 but not the leading twist distribution amplitudes that dominate the results.     

Two-site quantum random walk

We study the measure theory of a two-site quantum random walk. The truncated decoherence functional defines a quantum measure μ _n on the space of n-paths, and the μ _n in turn induce a quantum measure μ on the cylinder sets within the space Ω of untruncated paths. Although μ cannot be extended to a continuous quantum measure on the full σ-algebra generated by the cylinder sets, an important question is whether it can be extended to sufficiently many physically relevant subsets of Ω in a systematic way. We begin an investigation of this problem by showing that μ can be extended to a quantum measure on a “quadratic algebra” of subsets of Ω that properly contains the cylinder sets. We also present a new characterization of the quantum integral on the n-path space.     

Scenario of Inflationary Cosmology from the Phenomenological Λ Models

Choosing the three phenomenological models of the dynamical cosmological term Λ, viz., , and Λ∼ρ where a is the cosmic scale factor, it has been shown by the method of numerical analysis for the considered non-linear differential equations that the three models are equivalent for the flat Universe k=0 and for arbitrary non-linear equation of state. The evolution plots for dynamical cosmological term Λ vs. time t and also the cosmic scale factor a vs. t are drawn here for k=0,+1. A qualitative analysis has been made from the plots which supports the idea of inflation and hence expanding Universe.     

Box Dimensions and Topological Pressure for some Expanding Maps

We consider an expanding map defined on an open region of the plane and study the box dimensions of its invariant sets. Under the condition that the map leaves invariant a “strong unstable foliation”, we prove that the box dimension of an invariant set is given by δ_F+δ_T, where δ_T is its dimension transverse to and δ_F is the root of a certain function involving topological pressure. Received: 6 November 1996 / Accepted: 16 May 1997     

On the dimension of deterministic and random Cantor-like sets,symbolic dynamics,and the Eckmann-Ruelle Conjecture

In this paper we unify and extend many of the known results on the dimension of deterministic and random Cantor-like sets in ? ⁿ , and apply these results to study some problems in dynamical systems. In particular, we verify the Eckmann-Ruelle Conjecture for equilibrium measures for Hölder continuous conformal expanding maps and conformal Axiom A^# (topologically hyperbolic) homeomorphims. We also construct a Hölder continuous Axiom A^# homeomorphism of positive topological entropy for which the unique measure of maximal entropy is ergodic and has different upper and lower pointwise dimensions almost everywhere. this example shows that the non-conformal Hölder continuous version of the Eckmann-Ruelle Conjecture is false. The Cantor-like sets we consider are defined by geometric constructions of different types. The vast majority of geometric constructions studied in the literature are generated by a finite collection ofp maps which are either contractions or similarities and are modeled by the full shift onp symbols (or at most a subshift of finite type). In this paper we consider much more general classes of geometric constructions: the placement of the basic sets at each step of the construction can be arbitrary, and they need not be disjoint. Moreover, our constructions are modeled by arbitrary symbolic dynamical systems. The importance of this is to reveal the close and nontrivial relations between the statistical mechanics (and especially the absence of phase transitions) of the symbolic dynamical system underlying the geometric construction and the dimension of its limit set. This has not been previously observed since no phase transitions can occur for subshifts of finite type. We also consider nonstationary constructions, random constructions (determined by an arbitrary ergodic stationary distribution), and combinations of the above.     

On mass corrections to the decay <Emphasis Type="Italic">P</Emphasis> → <Emphasis Type="Italic">l</Emphasis><Superscript>+</Superscript><Emphasis Type="Italic">l</Emphasis><Superscript>−</Superscript>

The Mellin-Barnes representation is used to improve the theoretical estimate of mass corrections to the width of a light pseudoscalar meson decay into a lepton pair, P → l ⁺ l ⁻. The full resummation of the terms ln(m _l ²/Λ²)(m _l ²/Λ²) ⁿ and (m _l ²/Λ²) ⁿ to the decay amplitude is performed, where m _l is the lepton mass and Λ ≈ m _ρ is the characteristic scale of the P → γ*γ* form factor. The total effect of the mass corrections for the e ⁺ e ⁻ channel is negligible and, for the μ⁺μ⁻channel, its order is of a few percent. The text was submitted by the authors in English.     

Bulk Viscous Bianchi Type V Cosmological Models with Decaying Cosmological Term Λ

We present bulk viscous Bianchi type V cosmological models with time-dependent cosmological term Λ. Exact solutions of Einstein field equations have been obtained by assuming shear scalar σ proportional to volume expansion θ. The coefficient of bulk viscosity is taken to be power function of energy density ρ or volume expansion θ. In these models cosmological term Λ come out to be negative. It is found that models obtained are expanding, shearing and non-rotating. They do not approach isotropy for large values of time t. Some observational parameters for the model have also been discussed.     

A dissipative network model with neighboring activation

We study the properties of spectrum and eigenstates of the Google matrix of a directed network formed by the procedure calls in the Linux Kernel. Our results obtained for various versions of the Linux Kernel show that the spectrum is characterized by the fractal Weyl law established recently for systems of quantum chaotic scattering and the Perron-Frobenius operators of dynamical maps. The fractal Weyl exponent is found to be ν ≈ 0.65 that corresponds to the fractal dimension of the network d ≈ 1.3. An independent computation of the fractal dimension by the cluster growing method, generalized for directed networks, gives a close value d ≈ 1.4. The eigenmodes of the Google matrix of Linux Kernel are localized on certain principal nodes. We argue that the fractal Weyl law should be generic for directed networks with the fractal dimension d < 2.     

Perfect fluid bianchi Type-I cosmological models with time varying <Emphasis Type="Italic">G</Emphasis> and Λ

Bianchi Type-I cosmological models containing perfect fluid with time varying G and Λ have been presented. The solutions obtained represent an expansion scalar θ bearing a constant ratio to the anisotropy in the direction of space-like unit vector λ ⁱ . Of the two models obtained, one has negative vacuum energy density, which decays numerically. In this model, we obtain Λ ∼ H ², Λ ∼ R ₄₄/R and Λ ∼ T ⁻² (T is the cosmic time) which is in accordance with the main dynamical laws for the decay of Λ. The second model reduces to a static solution with repulsive gravity.      

First measurements of dtμ cycle characteristics in liquid H/D/T mixture

Muon catalyzed fusion in a dense triple mixture of hydrogen isotopes was investigated for the first time. The experimental method is based on the registration of neutrons from dtμ fusion by a full absorption detector in 4π geometry. The measurements were performed in H/D/T mixture at T=21 K and φ≃ 1.1 LHD using four sets of isotope concentrations. The basic parameters of the dtμ cycle (neutron yield, cycling rate and total sticking) in H/D/T mixtures obtained from neutron time distributions are presented and discussed. This revised version was published online in August 2006 with corrections to the Cover Date.