Dimension spectrum of axiom a diffeomorphisms. I. The Bowen-Margulis measure

We compute the dimension spectrumf() of the singularity sets of the Bowen-Margulis measure defined on a two-dimensional compact manifold and invariant with respect to aC ² Axiom A diffeomorphism. It is proved thatf is the Legendre-Fenchel transform of a free energy function which is real analytic (linear in the degenerate case). The functionf is also real analytic on its definition domain (defined in one point in the degenerate case) and is related to the Hausdorff dimensions of Gibbs measures singular with respect to each other and whose supports are the singularity sets, and we decompose these sets.     

Hyperbolicity,sinks and measure in one dimensional dynamics

Letf be aC ² map of the circle or the interval and let(f) denote the complement of the basins of attraction of the attracting periodic orbits. We prove that(f) is a hyperbolic expanding set if (and obviously only if) every periodic point is hyperbolic and(f) doesn't contain the critical point. This is the real one dimensional version of Fatou's hyperbolicity criteria for holomorphic endomorphisms of the Riemann sphere. We also explore other applications of the techniques used for the result above, proving, for instance, that for everyC ² immersionf of the circle (i.e. a map of the circle onto itself without critical points), either its Julia set has measure zero or it is the whole circle and thenf is ergodic, i.e. positively invariant Borel sets have zero or full measure.     

The dimension spectrum of some dynamical systems

We analyze the dimension spectrum previously introduced and measured experimentally by Jensen, Kadanoff, and Libchaber. Using large-deviation theory, we prove, for some invariant measures of expanding Markov maps, that the Hausdorff dimensionf() of the set on which the measure has a singularity is a well-defined, concave, and regular function. In particular, we show that this is the case for the accumulation of period doubling and critical mappings of the circle with golden rotation number. We also show in these particular cases that the functionf is universal.     

Absolutely continuous invariant measures for one-parameter families of one-dimensional maps

Given a one-parameter familyf (x) of maps of the interval [0, 1], we consider the set of parameter values for whichf has an invariant measure absolutely continuous with respect to Lebesgue measure. We show that this set has positive measure, for two classes of maps: i)f (x)=f(x) where 0<4 andf(x) is a functionC ³-near the quadratic mapx(1–x), and ii)f (x)=f(x) (mod 1) wheref isC ³,f(0)=f(1)=0 andf has a unique nondegenerate critical point in [0, 1].     

Spherical harmonics and cartesian tensor scalar product expansions of the distribution function

On the basis of the expansion of the distribution function in a sum of the spherical harmonics, the distribution functionf(v, r, t) is expanded in a series of scalar products of two Cartesian tensors term by term, i.e. The tensors and ^(l) (l=2, 3) are constructed in dependence on the spherical harmonic expansion coefficients (the tensors and ^(l) (l=0, 1) have been constructed by Jancel and Kahan [3]). On the basis of the knowledge of the analytic form off ₂ andf ₃ the equations forf ₁ f ₂ andf ₃ for the case of the Boltzmann's equation are determined.Technická 2, Praha 6, Czechoslovakia.     

The thermodynamic formalism for expanding maps

Letf:XX be an expanding map of a compact space (small distances are increased by a factor >1). A generating function(z) is defined which countsf-periodic points with a weight. One can express in terms of nonstandard Fredholm determinants of certain transfer operators, which can be studied by methods borrowed from statistical mechanics. In this paper we review the spectral properties of the transfer operators and the corresponding analytic properties of(z). Gibbs distributions and applications to Julia sets are also discussed. Some new results are proved, and some natural conjectures are proposed.This is an expanded version of the Bowen lectures given by the author at U.C. Berkeley in November 1988     

Schrödinger equation as an operator acting on test functions describing observing systems and the EPR correlation

A reformulation of quantum mechanics is introduced by regarding the Schrödinger equationE(f ⁺) = 0 for the retarded particle wavef ⁺ as an operator (functional) acting on the test functionf ^– satisfying the boundary conditions of the observing system: E(f ⁺),f ^– = 0. The variational expression for the transition amplitude of a particle between the particle source and the detector naturally arises in the dual space of the particle field and the test function. In the two-slit electron interference experiment, the test function plays the role of the quantum potential which carries the information of the detector and the slit locations backwards in time, while in the Einstein-Podolsky-Rosen process the test function describes the time reversed process of a pair of spatially separated fermions with arbitrarily chosen spin orientations progressing backwards in time to form a spherically symmetric compound state. The separation of the kinematics (spin correlation and the dynamics (spacetime aspect) of the EPR process is pointed out.     

Mean of the singularities of a Gibbs measure

We calculate the value of teh average of the singularities of a Gibbs measure invariant with respect to an expansiveC ² diffeomorphism of a one-compact manifold. This is the value related to dimension that one computes numerically. We then define and study a function, known as the correlation dimension, which is related to a free energy function, and we generalize the results in higher dimension with an axiomA transformation acting on a two-compact manifold.     

Differential forms of the dispersion integral

We discuss the derivative analyticity relations which were originally proposed as an alternative to dispersion relations; the dispersion integral is replaced by a tangent series of derivatives of the functionf which is, in the majority of high-energy applications, the imaginary part of a scattering amplitude. We consider three ways how to give the tangent series precise meaning. If the series converges on a real interval,f must be extensible to an entire function. Iff C and the dispersion integral converges, the latter is equal to a generalized sum of the tangent series. Finally in the high-energy limit, derivative relations are valid in which the tangent series is replaced by its first term. Then the class of applicability includes the majority of physically interesting functions.Invited talk presented at the International Conference Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 23–27, 1986.We are indebted to I. Vrko for valuable comments. We would like to thank H. Leutwyler and M. Jacob for hospitality during our stays at the Institute for Theoretical Physics of the University of Bern and at the Theory Division of CERN, Geneva.     

Direction-Dependent Free Energy Singularity of the Antiferroelectric Asymmetric Six-Vertex Model

The transition from the ordered commensurate phase to the incommensurate Gaussian phase of the antiferroelectric asymmetric six-vertex model is investigated by keeping the temperature constant below the roughening point and varying the external fields (h, v). In the (h, v) plane, the phase boundary is approached along straight lines v = k h, where (h, v) measures the displacement from the phase boundary. It is found that the free energy singularity displays the exponent 3/2 typical of the Pokrovski–Talapov transition f const(h)^3/2 for any direction other than the tangential one. In the latter case f shows a discontinuity in the third derivative.     

Stability of naked singularity arising in gravitational collapse of Type I matter fields

Considering gravitational collapse of Type I matter fields, we prove that, given an arbitrary C²-mass functionM(r, v) and a C¹-functionh(r, v) (through the corresponding C¹-metric functionν(t, r)), there exist infinitely many choices of energy distribution functionb(r) such that the ’true’ initial data(M, h(r,v)) leads the collapse to the formation of naked singularity. We further prove that the occurrence of such a naked singularity is stable with respect to small changes in the initial data. We remark that though the initial data leading to both black hole (BH) and naked singularity (NS) form a ’big’ subset of the true initial data set, their occurrence is not generic. The terms ’stability’ and ’genericity’ are appropriately defined following the theory of dynamical systems. The particular case of radial pressurep _r (r) has been illustrated in details to get a clear picture of how naked singularity is formed and how, it is stable with respect to initial data.     

Ising Models on Hyperbolic Graphs II

We consider Ising models with ferromagnetic interactions and zero external magnetic field on the hyperbolic graph (v, f), where v is the number of neighbors of each vertex and f is the number of sides of each face. Let T _c be the critical temperature and T _c =supTT _c: ^f=( ⁺+ ^–)/2, where ^f is the free boundary condition (b.c.) Gibbs state, ⁺ is the plus b.c. Gibbs state and ^– is the minus b.c. Gibbs state. We prove that if the hyperbolic graph is self-dual (i.e., v=f) or if v is sufficiently large (how large depends on f, e.g., v35 suffices for any f3 and v17 suffices for any f17) then 0<T _c <T _c, in contrast with that T _c =T _c for Ising models on the hypercubic lattice Z ^d with d2, a result due to Lebowitz.⁽²²⁾ While whenever T<T _c , ^f=( ⁺+ ^–)/2. The last result is an improvement in comparison with the analogous statement in refs. 28 and 33, in which it was only proved that ^f=( ⁺+ ^–)/2 when TT _c and it remains to show in both papers that ^f =( ⁺+ ^–)/2 whenever T<T _c . Therefore T _c and T _c divide [0, ] into three intervals: [0, T _c ), (T _c , T _c), and (T _c, ] in which ^{+ –} but ^f =( ⁺+ ^–)/2, ^{+ –} and ^f ( ⁺+ ^–)/2, and ⁺= ^–, respectively.     

On a regularity problem occurring in connection with Anosov diffeomorphisms

Let be aC -manifold and _s and _u be two Hölder foliations, transverse, and with uniformlyC leaves. If a functionf is uniformlyC along the leaves of the two foliations, then it isC on . The proof is elementary.     

Spectral asymptotics for the Schrödinger operator with potential which steadies at infinity

We consider the discrete spectrum of the selfadjoint Schrödinger operatorA _h =–h ² +V defined inL ²(^m) with potentialV which steadies at infinity, i.e.V(x)=g+|x|^– f(1+o(1)) as |x| for>0 and some homogeneous functionsg andf of order zero. Let _h (),0, be the total multiplicity of the eigenvalues ofA _h smaller thanM–, M being the minimum value ofg over the unit sphereS ^m–1 (hence,M coincides with the lower bound of the essential spectrum ofA _h ). We study the asymptotic behaviour of ₁() as0, or of _h () ash0, the number0 being fixed. We find that these asymptotics depend essentially on the structure of the submanifold ofS ^m–1, where the functiong takes the valueM, and generically are nonclassical, i.e. even as a first approximation (2) ^m _h () differs from the volume of the set {(x, )^2m:h ²||²+V(x)<M–}.Partially supported by Contract No. 52 with the Ministry of Culture, Science and Education     

Extensions of the Curzon metric

By examining the behaviour of geodesics approaching the singularity of the Curzon solution, it is shown that the metric is capable of being extended in such a way that almost all such geodesics are complete. There are an infinite number of possible extensions. None are analytic, but all areC .     

Endomorphisms of the Separated Product of Lattices

To describe the evolution of separated entities remaining separated, we proposeto study endomorphisms (join-preserving maps, sending atoms to atoms) of theseparated product of cao lattices (complete, atomistic orthocomplementedlattices). Morphisms have been used successfully to describe the evolution ofentities, and the separated product is a model for the property lattice of separatedsystems; its set of atoms is the Cartesian product of each atom space. Let L bethe separated product of two cao lattices having the covering property and f anendomorphism of L. We prove that the center F(L) of L is the power set of₁ × ₂ where _i is the atom space ofF(L _i ) (Theorem 1), f preserves irreduciblecomponents (Theorem 2), and if L is irreducible there exist two endomorphismsf ¹ and f ² and a permutation such that the restriction of f to atoms is given byf(p ₁, p ₂) = (f ¹(p ₍₁₎), f ²(p ₍₂₎))(Theorem 3). For generalizations of these resultsto separated products of families of cao lattices, we develop new general argumentsinvolving a topology we define on the set of atoms of a cao lattice.     

A Novel Series Solution to the Renormalization-Group Equation in QCD

Recently, the QCD renormalization-group (RG) equation at higher orders in MS-like renormalization schemes has been solved for the running coupling as a series expansion in powers of the exact two-loop-order coupling. In this work, we prove that the power series converge to all orders in perturbation theory. Solving the RG equation at higher orders, we determine the running coupling as an implicit function of the two-loop-order running coupling. Then we analyze the singularity structure of the higher-order coupling in the complex two-loop coupling plane. This enables us to calculate the radii of convergence of the series solutions at the three- and four-loop orders as a function of the number of quark flavours n _f . In parallel, we discuss in some detail the singularity structure of the coupling at the three- and four-loops in the complex-momentum squared plane for 0 ≤ n _f ≤ 16. The correspondence between the singularity structure of the running coupling in the complex-momentum squared plane and the convergence radius of the series solution is established. For sufficiently large n _f values, we find that the series converges for all values of the momentum-squared variable Q ² = −q ² > 0. For lower values of n _f , in the scheme, we determine the minimal value of the momentum-squared Q _min ² above which the series converges. We study properties of the non-power series corresponding to the presented power-series solution in the QCD analytic perturbation-theory approach of Shirkov and Solovtsov. The Euclidean and Minkowskian versions of the non-power series are found to be uniformly convergent over the whole ranges of the corresponding momentum-squared variables.     

Localization for random Schrödinger operators with correlated potentials

We prove localization at high disorder or low energy for lattice Schrödinger operators with random potentials whose values at different lattice sites are correlated over large distances. The class of admissible random potentials for our multiscale analysis includes potentials with a stationary Gaussian distribution whose covariance functionC(x,y) decays as |x–y|^–, where >0 can be arbitrarily small, and potentials whose probability distribution is a completely analytical Gibbs measure. The result for Gaussian potentials depends on a multivariable form of Nelson's best possible hypercontractive estimate.Partially supported by the NSF under grant PHY8515288Partially supported by the NSF under grant DMS8905627     

On the fundamentals of extended thermodynamics (ET) of a one-dimensional rarefied gas

Extended thermodynamics (ET) of degreer for a one-dimensional rarefied gas based, by definition, on a finite set A^r={a⁰, a²,..., a^r} of the firstr–1(3r) direct internal moments of the one-point distribution functionf is carefully investigated. With the aid of the second axiom of thermodynamics, the new representation forf, depending in a local and nonlinear way onA ^r , is explicitly derived. It is demonstrated that in ET of degreer an infinite sequence {b^r+1, b^r+2,...} ofhigher order Hermite coefficients, which normally drops out of Grad's proposition forf fashioned by mathematical apparatus such as the Hermite polynomials, cannot be considered negligible in the case when nonlinear constitutive functions are established. Using Ma's kinetic equation corresponding to a one-dimensional rarefied gas as well as the generalized representation forf, collision productions in the nonconservative moment equations are then calculated for a special choice of the rate of collisions between particles.     

Nonuniversality and analytical continuation in moments of directed polymers on hierarchical lattices

We prove the moments of the directed polymer partition function GZ, using an exact position space renormalization group scheme on a hierarchical lattice. After sufficient iteration the characteristic functionf(n)=lnGZⁿ of the probability (Z) converges to a stable limitf ^*(n). For smalln the limiting behavior is independent of the initial distribution, while for largen,f ^*(n) is completely determined by it and is thus nonuniversal. There is a smooth crossover between the two regimes for small effective dimensions, and the nonlinear behavior of the small moments can be used to extract information on the universal scaling properties of the distribution. For large effective dimensions there is a sharp transition between the two regimes, and analytical continuation from integer moments ton0 is not possible. Replica arguments can account for most features of the observed results.