a linear code for the sawtooth and cat maps abstract the sawtooth maps are a one parameter set of piecewise linear area preserving maps on the torus for positive integer values of the parameter k they are automorphisms of the torus known as the cat maps we present a symbolic dynamics for these maps in which the symbols are integers this code is related to a practical problem of the stabilisation of a system which is perturbed by impulses the code is linear in the sense that an orbit and its code are linearly related so it is not difficult to obtain a good approximation to one from the other in practice a stationary stochastic process for generating the code is given explicitly the theory uses green function methods which are also used to study ordered periodic orbits and cantori the problems of using a similar code for arbitrary area preserving twist maps on the torus are briefly discussed

threshold to global diffusion in a nonmonotonic map with quadratic nonlinearity abstract in this paper the threshold to global diffusion of a nonmonotonic map with quadratic nonlinearity using a forcing function k sin as in the standard map is analyzed for low values of the stochastic parameter k the breaking of the last kam curve is caused by the period one reconnecting resonance at higher k secondary resonances play an increasingly important role two kinds of kam curves are studied those between reconnecting resonances and those outside a reconnecting resonance pair using a perturbative hamiltonian method to determine the resonance width a weak overlap criterion is used to estimate the breaking of the last kam curve outside of the reconnecting islands for some values of the parameters a nonintegrable reconnecting threshold is found above the threshold of global diffusion in this regime the reconnection increases the diffusion coefficient to a value close to the quasilinear value of the standard map

numerical investigations of the structure around an invariant kam torus using the frequency map analysis abstract the hierarchical structure around kam tori is studied numerically on the standard map as a model problem applying the laskar frequency map analysis the theoretical prediction recently given by morbidelli and giorgilli j stat phys 78 1994 about the existence of a neighborhood almost completely full of slave tori around a chief torus are recovered and extended to the near critical regime

random cluster dynamics in mathrm mathbb z 2 the random cluster model has been widely studied as a unifying framework for random graphs spin systems and electrical networks but its dynamics have so far largely resisted analysis in this paper we analyze the glauber dynamics of the random cluster model in the canonical case where the underlying graph is an n times n box in the cartesian lattice mathrm mathbb z 2 our main result is a o n 2 log n upper bound for the mixing time at all values of the model parameter p except the critical point p p c q and for all values of the second model parameter q ge 1 we also provide a matching lower bound proving that our result is tight our analysis takes as its starting point the recent breakthrough by beffara and duminil copin on the location of the random cluster phase transition in mathrm mathbb z 2 it is reminiscent of similar results for spin systems such as the ising and potts models but requires the reworking of several standard tools in the context of the random cluster model which is not a spin system in the usual sense

algorithmic pirogov sinai theory we develop an efficient algorithmic approach for approximate counting and sampling in the low temperature regime of a broad class of statistical physics models on finite subsets of the lattice mathbb z d and on the torus mathbb z n mathbb z d our approach is based on combining contour representations from pirogov sinai theory with barvinok s approach to approximate counting using truncated taylor series some consequences of our main results include an fptas for approximating the partition function of the hard core model at sufficiently high fugacity on subsets of mathbb z d with appropriate boundary conditions and an efficient sampling algorithm for the ferromagnetic potts model on the discrete torus mathbb z n mathbb z d at sufficiently low temperature

universal transition in a dynamical system forced at two incommensurate frequencies abstract the various routes to chaos are explored for a nonlinear mechanical system mode locked to one of two incommensurate external frequencies two types of transitions are seen in our model system a saddle node transition with its associated intermittency can occur when the mode locking is lost while the attracting two torus remains smooth a less trivial transition can occur in which the attracting torus roughens and the power spectrum of the time series develops singular low frequency components after the breakdown mode locking persists with noisy small scale motions about the former torus at the multicritical point where these two transition lines meet scaling and universal low frequency power spectra are observed a renormalization group treatment is proposed the analysis might also be applicable to transitions in rotationally invariant systems

modular smoothing and kam tori abstract transformation properties of kam tori under the action of a modular group on frequencies are analysed and used for calculating the kam tori of a hamiltonian system with one and a half degrees of freedom first we derive a relation between the kam torus with a frequency v and the kam torus with the frequency v 1 v 1 v where 1 v is the integer part of 1 v given by a continuous function l1 v in the double limit v m n and k 0 where m n is a rational frequency and k is a perturbation parameter the function l1 v was introduced by buric percival and vivaldi and also gives the transformation properties of the critical parameter as a function of the frequency this function provides a zeroth order modular smoothing approximation to the invariant tori which is better than a perturbation expansion the first order corrections to this approximation can be calculated from a function r1 v that appears to be smooth

the regularity chaos regularity transition in a periodically driven anharmonic oscillator abstract a complete analytical and numerical investigation of the classical phase space structure for monochromatically driven x n anharmonic oscillator is presented the existence of additional chaos to regularity transition for a family of considered hamiltonian systems have been found analytically numerical calculations of the critical amplitude of the external field confirm the theory

on weak mixing in lattice models for lattice models on d weak mixing is the property that the influence of the boundary condition on a finite decays exponentially with distance from that region for a wide class of models on 2 including all finite range models we show that weak mixing is a consequence of gibbs uniqueness exponential decay of an appropriate form of connectivity and a natural coupling property in particular on 2 the fortuin kasteleyn random cluster model is weak mixing whenever uniqueness holds and the connectivity decays exponentially and the q state potts model above the critical temperature is weak mixing whenever correlations decay exponentially a hypothesis satisfied if q is sufficiently large ratio weak mixing is the property that uniformly over events a and b occurring on subsets and respectively of the lattice p a b p a p b 1 decreases exponentially in the distance between and we show that under mild hypotheses for example finite range weak mixing implies ratio weak mixing

exact results for an approximate renormalisation scheme and some predictions for the breakup of invariant ori abstract an approximate renormalisation scheme is derived for the breakup of invariant tori of arbitrary winding ratio in hamiltonian systems of one and a half degrees of freedom similar to that of escande and doveil it is a free semi group with two generators this scheme is solved exactly for its orbits stable manifolds unstable manifolds and critical set various results are found including a cantor set of universal fractal diagrams the robustness of noble tori and a scaling law for areas near critical circles

universal properties of the transition from quasi periodicity to chaos in dissipative systems abstract an exact renormalization group transformation is developed for dissipative systems which describes how the transition to chaos may occur in a continuous and universal manner if the frequency ratio in the quasi periodic regime is held at a fixed irrational value our approach is a natural extension of k a m theory to strong coupling most of our analysis is for analytic circle maps we have found a strong coupling fixed point where invertibility is lost which describes the universal features of the transition to chaos we find numerically that any two such critical maps with the same winding number are c 1 conjugate it follows that the low frequency peaks in an experimental spectrum are universal and we determine how their envelope scales with frequency when the winding number has a periodic continued fraction our renormalization transform has a fixed point and spectra are self similar in addition for a set of non periodic winding numbers with full measure our renormalization transformation yields an ergodic trajectory in a sub space of all critical maps physically one finds singular and universal spectra that do not scale

natural boundaries for area preserving twist maps we consider kam invariant curves for generalizations of the standard map of the form x y x y y f x wheref x is an odd trigonometric polynomial we study numerically their analytic properties by a pade approximant method applied to the function which conjugates the dynamics to a rotation in the complex plane natural boundaries of different shapes are found in the complex plane the analyticity region appears to be a strip bounded by a natural boundary whose width tends linearly to 0 as tends to the critical value

renormalization method for computing the threshold of the large scale stochastic instability in two degrees of freedom hamiltonian systems an approximate renormalization procedure is derived for the hamiltonianh v x t v2 2 m cosx p cosk x t it gives an estimate of the large scale stochastic instability threshold which agrees within 5 10 with the results obtained from direct numerical integration of the canonical equations it shows that this instability is related to the destruction of kam tori between the two resonances and makes the connection with kam theory possible improvements of the method are proposed the results obtained forh allow us to estimate the threshold for a large class of hamiltonian systems with two degrees of freedom

construction of analytic kam surfaces and effective stability bounds a class of analytic possibly time dependent hamiltonian systems withd degrees of freedom and the corresponding class of area preserving twist diffeomorphisms of the plane are considered implementing a recent scheme due to moser salamon and zehnder we provide a method that allows us to construct explicitly kam surfaces and hence to give lower bounds on their breakdown thresholds we then apply this method to the hamiltonianh y2 2 e cosx cos x t and to the map y x y e sinx x y e sinx obtaining with the aid of computer assisted estimations explicit approximations within an error of 10 5 of the golden mean kam surfaces for complex values of e with e less or equal than respectively 0 015 and 0 65 the experimental numerical values at which such surfaces are expected to disappear are about respectively 0 027 and 0 97 a possible connection between break down thresholds and singularities in the complex e plane is pointed out

destruction of kam tori in hamiltonian systems link with the destabilization of nearby cycles and calculation of residues abstract the disappearance of kam tori is studied on the non integrable hamiltonian h x v t v 2 2 m cos x p cos k x t a first method which is an approximate renormalization procedure allows one to make a local description of the phase space the stability of cycles is studied by means of a second method which involves a mathieu equation the compounding of the two methods allows one to compute analytically the residue r and the mean residue first introduced by j greene one recovers and analytically generalizes his main numerical conjectures about the link between kam tori and nearby cycles the existence of a mean residue is shown for almost all tori in the sense of the lebesgue measure critical residues are shown to be universal quantities in accordance with results of s j shenker and l p kadanoff

tight bounds for mixing of the swendsen wang algorithm at the potts transition point we study two widely used algorithms for the potts model on rectangular subsets of the hypercubic lattice mathbb z d heat bath dynamics and the swendsen wang algorithm and prove that under certain circumstances the mixing in these algorithms is torpid or slow in particular we show that for heat bath dynamics throughout the region of phase coexistence and for the swendsen wang algorithm at the transition point the mixing time in a box of side length l with periodic boundary conditions has upper and lower bounds which are exponential in ld 1 this work provides the first upper bound of this form for the swendsen wang algorithm and gives lower bounds for both algorithms which significantly improve the previous lower bounds that were exponential in l log l 2

rigorous estimates for a computer assisted kam theory nonautonomous hamiltonian systems of one degree of freedom close to integrable ones are considered let e be a positive parameter measuring the strength of the perturbation and denote by e c the critical value at which a given kam kolmogorov arnold moser torus breaks down a computer assisted method that allows one to give rigorous lower bounds for e c is presented this method has been applied in celletti falcolini porzio to be published in ann inst h poincare to the escande and doveil pendulum yielding a bound which is within a factor 40 2 of the value indicated by numerical experiments

on barnes beta distributions and applications to the maximum distribution of the 2d gaussian free field a new family of barnes beta distributions on 0 infty is introduced and its infinite divisibility moment determinacy scaling and factorization properties are established the morris integral probability distribution is constructed from barnes beta distributions of types 1 0 and 2 2 and its moment determinacy and involution invariance properties are established for application the maximum distributions of the 2d gaussian free field on the unit interval and circle with a non random logarithmic potential are conjecturally related to the critical selberg and morris integral probability distributions respectively and expressed in terms of sums of barnes beta distributions of types 1 0 and 2 2

a uniqueness condition for gibbs measures with application to the 2 dimensional ising antiferromagnet a uniqueness condition for gibbs measures is given this condition is stated in terms of absence of a certain type of percolation involving two independent realisations this result can be applied in certain concrete situations by comparison with ordinary percolation in this way we prove that the ising antiferromagnet on a square lattice has a unique gibbs measure if 4 h 1 2ln pc 1 pc whereh denotes the external magnetic field the inverse temperature andpc the critical probability for site percolation on that lattice sincepc is larger than 1 2 this extends a result by dobrushin kolafa and shlosman whose proof was computer assisted

exact thresholds for ising gibbs samplers on general graphs we establish tight results for rapid mixing of gibbs samplers for the ferromagnetic ising model on general graphs we show that if d 1 tanh 1 then there exists a constant c such that the discrete time mixing time of gibbs samplers for the ferromagnetic ising model on any graph of n vertices and maximal degree d where all interactions are bounded by and arbitrary external fields are bounded by cnlogn moreover the spectral gap is uniformly bounded away from 0 for all such graphs as well as for infinite graphs of maximal degree d we further show that when dtanh 1 with high probability over the erd s renyi random graph g n d n it holds that the mixing time of gibbs samplers is n1 1 loglogn both results are tight as it is known that the mixing time for random regular and erd s renyi random graphs is with high probability exponential in n when d 1 tanh 1 and dtanh 1 respectively to our knowledge our results give the first tight sufficient conditions for rapid mixing of spin systems on general graphs moreover our results are the first rigorous results establishing exact thresholds for dynamics on random graphs in terms of spatial thresholds on trees

on the analytic treatment of non integrable difference equations guided by general features of the solution of non integrable difference equations that emerge from kam theory a local perturbation theory is developed this captures both qualitatively and quantitatively all the essential features of the solution the method is applied to the standard map of chirikov 1979

a renormalization approach to invariant circles in area preserving maps abstract kadanoff and shenker introduced a renormalisation approach to invariant circles in area preserving maps this paper makes more precise the connection between invariant circles and the renormalisation operator restricting attention to noble rotation numbers the stability of a simple fixed point of the renormalisation is analysed corresponding to a linear twist map it is found to be essentially attracting so that noble circles persist under perturbation giving a new view on kam theory shenker and kadanoff found evidence for another fixed point corresponding to a map with a non smooth noble circle further evidence is given in this paper it has essentially only one unstable direction and its stable manifold is believed to give the boundary of the set of twist maps with a noble circle finally noble circles are shown to be locally most robust in an important sense

converse kam theory and practice we unify extend reinterpret and apply criteria of birkhoff 1 herman 9 mather 2 3 aubry et al 4 5 and newman and percival 6 for the nonexistence of invariant circles for area preserving twist maps the criteria enable one to establish regions of phase space through which no rotational invariant circles pass for families of maps the same can be done for regions of the combined space of phase points and parameters the criteria can be implemented rigorously on a computer and give a practical method of proving quite strong results as an example we present a computer program which proved that the standard map has no rotational invariant circles for any parameter value k 63 64

mixing properties and exponential decay for lattice systems in finite volumes an infinite volume mixing or exponential decay property in a spin system or percolation model reflects the inability of the influence of the configuration in one region to propagate to distant regions but in some circumstances where such properties hold propagation can nonetheless occur in finite volumes endowed with boundary conditions we establish the absense sic of such propagation particularly in two dimensions in finite volumes which are simply connected under a variety of conditions mainly for the potts model and the fortuin kasteleyn fk random cluster model allowing external fields for example for the fk model in two dimensions we show that exponential decay of connectivity in infinite volume implies exponential decay in simply connected finite volumes uniformly over all such volumes and all boundary conditions and implies a strong mixing property for such volumes with certain types of boundary conditions for the potts model in two dimensions we show that exponential decay of correlations in infinite volume implies a strong mixing property in simply connected finite volumes which includes exponential decay of correlations in simply connected finite volumes uniformly over all such volumes and all boundary conditions

critical ising on the square lattice mixes in polynomial time the ising model is widely regarded as the most studied model of spin systems in statistical physics the focus of this paper is its dynamic stochastic version the glauber dynamics introduced in 1963 and by now the most popular means of sampling the ising measure intensive study throughout the last three decades has yielded a rigorous understanding of the spectral gap of the dynamics on mathbb z 2 everywhere except at criticality while the critical behavior of the ising model has long been the focus for physicists mathematicians have only recently developed an understanding of its critical geometry with the advent of sle cle and new tools to study conformally invariant systems

the critical renormalization fixed point for commuting pairs of area preserving maps we prove the existence of the critical fixed point f g for mackay s renormalization operator for pairs of maps of the plane the maps f and g commute are area preserving reversible real analytic and they satisfy a twist condition

invariant curves for area preserving twist maps far from integrable one parameter families of area preserving twist maps of the formf e x y x y ef x y ef x are considered various invariant curves for the maps corresponding tof x sin andf x sinx 1 50 sin 5x are rigorously constructed forlarge values of the nonlinearity parameter for larger values of close to critical some numerical experiments are briefly discussed

renormalization for stochastic layers abstract chirikov s standard mapping for stochastic layers is derived in a simpler and more general way a one parameter renormalization transformation yields analytically many universal quantities defined in a previous renormalization theory expressions are given for the width of stochastic layers that correct several previous estimates

interfaces in the potts model i pirogov sinai theory of the fortuin kasteleyn representation we develop a new analysis of the order disorder transition in ferromagnetic potts models for large numberq of spin states we use the pirogov sinai theory which we adapt to the fortuin kasteleyn representation of the models this theory applies in a rather direct way in our approach and leads to a system of non interacting contours with small activities as a consequence simpler and more natural techniques are found allowing us to recover previous results on the bulk properties of the model which then extend to non integer values ofq and to deal with non translation invariant boundary conditions this will be applied in a second part of this work to study the behaviour of the interfaces at the transition point

the self dual point of the two dimensional random cluster model is critical for q 1 we prove a long standing conjecture on random cluster models namely that the critical point for such models with parameter q 1 on the square lattice is equal to the self dual point p sd q sqrt q 1 sqrt q this gives a proof that the critical temperature of the q state potts model is equal to log 1 sqrt q for all q 2 we further prove that the transition is sharp meaning that there is exponential decay of correlations in the sub critical phase the techniques of this paper are rigorous and valid for all q 1 in contrast to earlier methods valid only for certain given q the proof extends to the triangular and the hexagonal lattices as well

chaotic boundary of a hamiltonial map abstract the half plane maps are a class of complex hamiltonial maps whose invariant curves at most fixed irrational frequencies can be obtained as a convergent taylor series expansion for these maps the boundary between regular ordered motion on invariant curves and irregular chaotic motion is given by the radius of convergence of the series the successive terms of the series oscillate wildly due to the presence of small divisors methods are presented for taming the series based on the conversion of the convergenceexponentc ln c into the integral of a continuous but nondifferentiable lambda function whose graph whows a similarity structure on small scales self similarity properties are illustrated for the chaotic boundary function c where v is the frequency

external field induced chaos in an infinite square well potential abstract we study the motion of a particle in an infinite square well potential in the presence of a monochromatic external field the equations of motion of this system have a particularly simple structure compared to other driven nonlinear systems and yet the system exhibits a transition to chaotic behavior the critical amplitudes of the external field where breakdown of kam invariants occurs and large scale chaos sets in are computed by numerical experiment they are found to be in good agreement with predictions based on a renormalization group scheme

for 2 d lattice spin systems weak mixing implies strong mixing we prove that for finite range discrete spin systems on the two dimensional latticez2 the weak mixing condition which follows for instance from the dobrushin shlosman uniqueness condition for the gibbs state implies a stronger mixing property of the gibbs state similar to the dobrushin shlosman complete analyticity condition but restricted to all squares in the lattice or more generally to all sets multiple of a large enough square the key observation leading to the proof is that a change in the boundary conditions cannot propagate either in the bulk because of the weak mixing condition or along the boundary because it is one dimensional as a consequence we obtain for ferromagnetic ising type systems proofs that several nice properties hold arbitrarily close to the critical temperature these properties include the existence of a convergent cluster expansion and uniform boundedness of the logarithmic sobolev constant and rapid convergence to equilibrium of the associated glauber dynamics on nice subsets ofz2 including the full lattice

universal small scale structure near the boundary of siegel disks of arbitrary rotation number abstract we present numerical evidence that for analytic maps of the complex plane with a siegel disk of any irrational rotation number and a quadratic critical point on its boundary the small scale structure near the critical point can be described asymptotically by a universal two parameter family we propose an explanation in terms of an ergodic attractor for a renormalisation operator