test sets for context free languages and algebraic systems of equations over a free monoid it is shown that for every context free language l there effectively exists a test set f that is a finite subset f of l such that for any pair g h of morphisms g x h x for all x in f implies g x h x for all x in l this result was claimed earlier but a detailed correct proof is given here together with very recent results on systems of equations over a free monoid this result implies that every algebraic system of equations is equivalent to a finite subsystem

ising models julia sets and similarity of the maximal entropy measures we study the phase transition of ising models on diamondlike hierarchical lattices following an idea of lee and yang one can make an analytic continuation of free energy of this model to the complex temperature plane it is known that the migdal kadanoff renormalization group of this model is a rational endomorphism denoted byf of and that the singularities of the free energy lie on the julia setj f the aim of this paper is to prove that the free energy can be represented as the logarithmic potential of the maximal entropy measure onj f moreover using this representation we can show a close relationship between the critical exponent and local similarity of this measure

electrical circuits on mesoscopic sierpinski gaskets a deterministic network of impedances on the sierpinski gasket fractal is considered with particular regard to low generation or mesoscopic systems we study the fractal set of the resonances of the circuit and present calculations on frequency dependent systems the condition of an intermediate size can lead to an oscillatory scaling of the total impedance that disappears in the asymptotic limit several examples are given

propagation of moments and regularity for the 3 dimensional vlasov poisson system we prove that if the initial data has moments inv higher than three then the solution of vlasov poisson has also moments inv higher than three we deduce from this different regularity results on the local density the force field or the solution itself also we give a new uniqueness result and new regularity results for solutions satisfying only the energy andl bounds our proofs are based on a new representation formula and logarithmic estimates for the force field

a rescaling approach to two and three dimensional lattice spectra an exact real space rescaling transformation is used to calculate the spectral properties of tight binding hamiltonians with an arbitrary range of interaction of two and three dimensional lattices the intrinsic problem of the proliferation of initial short range interactions under rescaling is shown to be tractable numerically this method provides a direct way of obtaining the local green functions and local density of states singularities of the spectral functions are associated with fixed points or limit cycles of the rescaling transformation band edge fixed points and exponents have been found for the square triangular and cubic lattices in two and three dimensions there is rapid convergence to accurate numerical results for the green function outside the band while good results have been obtained for the in band e g density of states spectra in two dimensions by suitable choices of cluster size and attenuation factor insight into the relationship between these factors follows from an investigation of finite cluster approximations to the exact transformation it is found that the zeros of the green functions in the complex plane are julia sets which have a fractal structure for a given cluster size the attenuation factors must lie outside these zeros in order for the procedure to be convergent

locally self similar phase diagram of the disordered potts model on the hierarchical lattice we study the critical behavior of the random q state potts model in the large q limit on the diamond hierarchical lattice with an effective dimensionality d eff 2 by varying the temperature and the strength of the frustration the system has a phase transition line between the paramagnetic and the ferromagnetic phases which is controlled by four different fixed points according to our renormalization group study the phase boundary in the vicinity of the multicritical point is self similar it is well represented by a logarithmic spiral we expect an infinite number of reentrances in the thermodynamic limit consequently one cannot define standard thermodynamic phases in this region

asymptotics of the susceptibility for the ising model on the hierarchical lattices using the gks inequality griffiths and kaufman showed in 12 that the susceptibility in the ising model on the hierarchical lattices is infinite fort tc in the present work we give an analytic calculation of the dominant singularity of the susceptibility whenh 0 fort tc moreover we analytically find the spontaneous magnetization fort tc and prove that fort tc the susceptibility is finite

on the lagrangian structure of transport equations the vlasov poisson system the vlasov poisson system is an important nonlinear transport equation used to describe the evolution of particles under their self consistent electric or gravitational field the existence of classical solutions is limited to dimensions d 3 under strong assumptions on the initial data whereas weak solutions are known to exist under milder conditions however in the setting of weak solutions it is unclear whether the eulerian description provided by the equation physically corresponds to a lagrangian evolution of the particles in this article we develop several general tools concerning the lagrangian structure of transport equations with nonsmooth vector fields and we apply these results to show that weak renormalized solutions of vlasov poisson are lagrangian and actually that the concepts of renormalized and lagrangian solutions are equivalent as a corollary we prove that finite energy solutions in dimension d 4 are transported by a global flow in particular they preserve all the natural casimir invariants and we obtain the global existence of weak solutions in any dimension under minimal assumptions on the initial data

one dimensional mapping modulated phases and lyapunov exponent for the antiferromagnetic q state potts and multi site exchange interaction ising models abstract we describe the bifurcation structure period doubling and chaos for the antiferromagnetic q state potts model on the bethe lattice and three site interaction ising model on husimi one in a magnetic field by using the recursion relation technique a chaotic behavior of the magnetic susceptibility for the models is observed at low temperatures the resulting one dimensional rational mapping has a positive lyapunov exponent in the region of the chaotic regime for the antiferromagnetic q state potts q 2 and three site interaction ising models we discuss modulated phases for the antiferromagnetic q state potts q 2 and q 2 and three site interaction ising model at low temperatures the q state potts model q 2 has only one modulated phase with 1 2 pinching corresponding to the 2 cycle the q state potts q 2 and three site interaction ising models have an infinite number of modulated phases with different pinching numbers we construct the first modulated phase after the first bifurcation point

physical meaning for mandelbrot and julia sets abstract we consider the dynamics of a kicked charged particle moving in a double well potential and a time dependent magnetic field in certain cases the stroboscopic dynamics reduces to the complex logistic map thus providing physical meaning for the mandelbrot set in other cases we obtain iterated function systems consisting of the inverse complex logistic map thus providing physical meaning for julia sets our approach can be generalized to complex mappings with a maximum of order q

the vlasov equation with infinite mass we show existence and uniqueness for the solutions to the vlasov equation for a positive smooth and short range potential in the case of unbounded mass

ferromagnetic phase transitions of inhomogeneous systems modelled by square ising models with diamond type bond decorations abstract the two dimensional ising model defined on square lattices with diamond type bond decorations is employed to study the nature of the ferromagnetic phase transitions of inhomogeneous systems the model is studied analytically under the bond renormalization scheme for an n level decorated lattice the long range ordering occurs at the critical temperature given by the fitting function k b t c j n 1 6410 0 6281 exp 0 5857 n and the local ordering inside n level decorated bonds occurs at the temperature given by the fitting function k b t m j n 1 6410 0 8063 exp 0 7144 n the critical amplitude a sin g n of the logarithmic singularity in specific heat characterizes the width of the critical region and it varies with the decoration level n as a sin g n 0 2473 exp 0 3018 n obtained by fitting the numerical results the cross over from a finite decorated system to an infinite decorated system is not a smooth continuation for the case of infinite decorations the critical specific heat becomes a cusp with the height c n 0 639852 the results are compared with those obtained in the cell decorated ising model

fractal structure of zeros in hierarchical models we consider an ising and aq state potts model on a diamond hierarchical lattice we give pictures of the distribution of zeros of the partition function in the complex plane of temperatures for several choices ofq these zeros are just the julia set corresponding to the renormalization group transformation

meblin transforms associated with julia sets and physical applications we introduce the mellin transform of the balanced invariant measure associated to the julia set generated by a rational transformation we show that its analytic continuation is a meromorphic function the poles of which are on a semi infinite periodic lattice this allows one to have an understanding of the behavior of the measure near a repulsive fixed point trace identities corresponding to the fact that the analytically continued mellin transform vanishes at negative integers are derived for the polynomial case the quadratic map is first analyzed in detail and the analytic properties of the inverse of the green s function are exhibited of interest is the appearance of a dense set of spikes at dyadic points when the julia set is disconnected these results are used to study the residues of the mellin transform a certain number of physically interesting consequences are derived for the spectral dimensionality of quantum mechanical systems the excitation spectrum of which displays unusual oscillations the appearance of complex critical indices for thermodynamical systems is also discussed in the conclusion

geometry and combinatorics of julia sets of real quadratic maps for real a correspondence is made between the julia setb forz z 2 in the hyperbolic case and the set of chains with the aid of cremer s theorem it is shown how a number of features ofb can be understood in terms of chains the structure ofb is determined by certain equivalence classes of chains fixed by orders of visitation of certain real cycles and the bifurcation history of a given cycle can be conveniently computed via the combinatorics of chains the functional equations obeyed by attractive cycles are investigated and their relation to chains is given the first cascade of period doubling bifurcations is described from the point of view of the associated julia sets and chains certain julia sets associated with the feigenbaum function and some theorems of lanford are discussed

on the 3 d vlasov poisson system with point charges global solutions with unbounded supports and propagation of velocity spatial moments abstract a new energy functional which is shown to decay to zero as time tends to infinite is introduced for the three dimensional vlasov poisson plasma in the presence of heavy point charges with repulsive interaction moreover a new kind of pointwise energy of a plasma particle relative to each point charge is also constructed and the variation of its root is shown to be controlled by the electrostatic fields based on those results we prove global existence and uniqueness of a classical solution possibly having infinite kinetic energy but with compact velocity spatial support specified in theorem 1 1 furthermore for the single point charge vlasov poisson plasma global existence and polynomial propagation of velocity spatial moments of weak solutions are also established

log periodic critical amplitudes a perturbative approach log periodic amplitudes appear in the critical behavior of a large class of systems in particular when a discrete scale invariance is present here we show how to compute these critical amplitudes perturbatively when they originate from a renormalization map which is close to a monomial in this case the log periodic amplitudes of the subdominant corrections to the leading critical behavior can also be calculated

the vlasov poisson system with infinite kinetic energy and initial data in lp r6 abstract in this paper the global existence of weak solutions to the three dimensional vlasov poisson system with infinite kinetic energy and initial data in l p r 6 is studied the propagation of low moments in the velocity variables is also proved

julia sets and complex singularities in hierarchical ising models we study the analytical continuation in the complex plane of free energy of the ising model on diamond like hierarchical lattices it is known 12 13 that the singularities of free energy of this model lie on the julia set of some rational endomorphismf related to the action of the migdal kadanoff renorm group we study the asymptotics of free energy when temperature goes along hyperbolic geodesics to the boundary of an attractive basin off we prove that for almost all with respect to the harmonic measure geodesics the complex critical exponent is common and compute it

on systems of word equations with simple loop sets consider the infinite system s of word equations x 0u 1 ix 1u 2 ix 2 u m ix m y 0v 1 iy 1v 2 iy 2 v n iy n email protected n for each email protected n let t k be the subsystem of s given by email protected k k 1 k 2 we prove two properties of the above system 1 let k 1 if f is a solution of t k such that primitive roots of f u 1 f u 2 f u m are of equal length as well as primitive roots of f v 1 f v 2 f v n then f is a solution of the whole s 2 if n 1 then for any k 2 a solution f of t k is also a solution of s

time evolution of an infinitely extended vlasov system with singular mutual interaction we study the time evolution of an infinitely extended system in the mean field approximation governed by the vlasov equation this system is confined in an unbounded cylinder by an external force singular on the border the mutual interaction is assumed singular at short distance as 1 r alpha with alpha 2 3 or alpha 1 in case of an external lorentz force and with a short range the initial density is assumed bounded differently from studies which assume initial data compact in space and or in velocities here we consider a system having infinite mass and an exponential bound on the velocities according to the maxwell boltzmann law

bounds on the growth of the velocity support for the solutions of the vlasov poisson equation in a torus a bound on the growth of the velocity for the vlasov poisson equation in a torus is given in one and two dimensions the main tool used in the proof is a partition into fast and slow particles and the ergodic property of the free motion in a torus

non equilibrium dynamics of three dimensional infinite particle systems we show existence and uniqueness for the solutions to the newton equations relative to a system of infinitely many particles in the space interacting by means of a positive and short range potential the initial conditions are chosen in a set sufficiently large to be the support of any reasonable non equilibrium state we extend previous results in one and two dimensions obtained by lanford and by fritz and dobrushin respectively many years ago

the yang lee edge singularity for the ising model on two sierpinski fractal lattices we study the distribution of zeros of the partition function of the ferromagnetic ising model near the yang lee edge on two sierpiski type lattices we have shown that relevant correlation length displays a logarithmic divergence near the edge where is a constant and h distance from the edge in the case of a modified sierpinski gasket with a nonuniform coordination number it is demonstrated that this critical behavior can be related to the critical behavior of a simple zero field gaussian model of the same structure we have shown that there is no such connection between these two models on a second lattice that has a uniform coordination number these findings suggest that fluctuations of the lattice coordination number of a nonhomogeneous self similar structure exert the crucial influence on the critical behavior of both models

a proof of ehrenfeucht s conjecture abstract ehrenfeucht s conjecture states that each subset s of a finitely generated free monoid has a finite subset t such that if two endomorphisms of the monoid agree on t then they agree on s it is the purpose of this note to verify the conjecture

renormalization group evaluation of exponents in family name distributions abstract according to many phenomenological and theoretical studies the distribution of family name frequencies in a population can be asymptotically described by a power law we show that the galton watson process corresponding to the dynamics of a growing population can be represented in hilbert space and its time evolution may be analyzed by renormalization group techniques thus explaining the origin of the power law and establishing the connection between its exponent and the ratio between the population growth and the name production rates

continuity of julia set and its hausdorff dimension of yang lee zeros abstract we show the continuity of the julia set and its hausdorff dimension about a family of rational maps concerning 2 dimensional diamond hierarchical potts models about anti ferromagnetic coupling in statistical mechanics

on the system of word equations x o u i i x 1 u i 2 x 2 u i 3 x 3 y 0 v i 1 y 1 v i 2 y 2 v i 3 y 3 i 0 1 2 l in a free monoid abstract we prove that if x 0 u 1 i x 1 u 2 i x 2 y 0 v 1 i y 1 v 2 i y 2 holds for all i 0 1 2 3 then it is true for each i 0 1 2 and that if x 0 u 1 i x 1 u 2 i x 2 u 3 i x 3 y 0 v 1 i y 1 v 2 i y 2 v 3 i y 3 holds for all i 0 1 2 3 4 5 then it is true for each i 0 1 2

mapping between hierarchical lattices by renormalisation and duality the renormalisation group scheme of migdal 1976 and kadanoff 1976 is applied to a ferromagnetic potts model on two hierarchical lattices in the two cases the renormalisation transformation is split into two elementary processes parallel bonds or series bonds and each lattice can be mapped onto the other one it is then easy to deduce the characteristic parameters critical exponents repulsive fixed points etc of one lattice from those of the other lattice moreover the classical geometrical duality offers a third possibility of mapping and the relationship between the renormalisation and the duality methods is presented in the peculiar case of a two dimensional lattice

a real space renormalisation group approach to electrical and noise properties of percolation clusters the migdal kadanoff renormalisation scheme which becomes exact on hierarchical lattices is used to investigate electrical properties of 2d and 3d percolation clusters the author studies in detail the frequency dependence of the conductivity and the loss angle and the amplification of resistance flicker noise this type of noise exhibits critical amplification with its own exponents x and y which are related to the exponent b recently defined by rammal et al 1985

the screened poisson vlasov system with infinite charge and vanishing screening the vlasov equation may be used to model the dynamics of a collisionless plasma we consider a situation where the plasma has unbounded support and the charge density does not decay as x the electric field is computed from the screened poisson equation whose fundamental solution decays exponentially as x and hence gives a convergent integral despite the lack of decay of the charge density the limit as the screening is removed is studied in one space dimension it is shown that if some assumptions on the initial conditions hold then the limit in question exists but is not uniform and the problem satisfied by the limit is identified

renormalisation group study of a disordered model we study the potts model on a diamond hierarchical lattice with random interactions using weak disorder expansions we calculate analytically the position and the exponents of the random fixed point which appears when the specific heat exponent ap of the pure system becomes positive at ap 0 we find how the logarithmic singularity is modified by the disorder lastly we suggest that this model should present griffiths like singularities