rational surface maps with invariant meromorphic two forms let f s dashrightarrow s be a rational self map of a smooth complex projective surface s and eta be a meromorphic two form on s satisfying f eta delta eta for some delta in mathbf c we show that under a mild topological assumption on f there is a birational change of domain psi x dashrightarrow s such that psi eta has no zeros in this context we investigate the notion of algebraic stability for f proving that f can be made algebraically stable if and only if it acts nicely on the poles of eta we illustrate this last result in the case eta frac dx wedge dy xy where we translate our stability result into a condition on whether a circle homeomorphism associated to f has rational rotation number

fast multipole method for scattering from an arbitrary pec target above or buried in a lossy half space the fast multipole method fmm was originally developed for perfect electric conductors pecs in free space through exploitation of the spectral properties of the free space green s function in the work reported here the fmm is modified for scattering from an arbitrary three dimensional 3 d pec target above or buried in a lossy half space the near terms in the fmm are handled via the original method of moments mom analysis wherein the half space green s function is evaluated efficiently and rigorously through application of the method of complex images the far fmm interactions which employ a clustering of expansion and testing functions utilize an approximation to the green s function dyadic via real image sources and far field reflection dyadics the half space fmm algorithm is validated through comparison with results computed via a rigorous mom analysis further a detailed comparison is performed on the memory and computational requirements of the mom and fmm algorithms for a target in the vicinity of a half space interface

the dirichlet problem for nonlinear second order elliptic equations iii functions of the eigenvalues of the hessian on etudie le probleme de dirichlet dans un domaine borne de r n a frontiere lisse f d 2 u dans u sur

efficient generation of method of moments matrices using the characteristic function method we present a new fast and efficient technique for computing the mom matrix elements from rao wilton glisson rwg bases one that does not require evaluating the usual surface integrals involving the expansion and testing functions the matrix elements are represented in terms of characteristic functions cfs which can be expressed in a closed canonical form these universal cfs make matrix generation more efficient and faster than the conventional triangular facet interaction scheme the proposed technique can be applied for any subsectional basis functions other the rwg if the size of the basis function is small enough compared to the wavelength and the distance between the source and the testing basis functions is beyond a nominal distance the proposed approach is validated for several canonical geometries and the accuracy of the generated matrix elements as well as the radar cross section rcs has been verified

polarized endomorphisms of complex normal varieties it is shown that a complex normal projective variety has non positive kodaira dimension if it admits a non isomorphic quasi polarized endomorphism the geometric structure of the variety is described by methods of equivariant lifting and fibrations

a study of wavelets for the solution of electromagnetic integral equations the use of wavelet basis functions for the efficient solution of electromagnetic integral equations is studied it has previously been demonstrated that the use of wavelets for expansion and testing functions produces a sparse moment method matrix here this effect is examined and analyzed in terms of the radiation receiving characteristics of the wavelet basis functions the limitations of wavelets as an efficient solution technique are discussed and a comparison is made to other fast algorithms

adaptive multiscale moment method ammm for analysis of scattering from perfectly conducting plates adaptive multiscale moment method ammm is presented for the analysis of scattering from a thin perfectly conducting plate this algorithm employs the conventional moment method and a special matrix transformation which is derived from the tensor products of the two one dimensional 1 d multiscale triangular basis functions that are used for expansion and testing functions in the conventional moment method the special feature of these new basis functions introduced through this transformation is that they are orthogonal at the same scale except at the initial scale and not between scales from one scale to another scale the initial estimate for the solution can be predicted using this multiscale technique hence the compression is applied directly to the solution and the size of the linear equations to be solved is reduced thereby improving the efficiency of the conventional moment method the basic difference between this methodology and the other techniques that have been presented so far is that we apply the compression not to the impedance matrix but to the solution itself directly using an iterative solution methodology the extrapolated results at the higher scale thus provide a good initial guess for the iterative method typically when the number of unknowns exceeds a few thousand unknowns the matrix solution time exceeds generally the matrix fill time hence the goal of this method is directed in solving electrically larger problems where the matrix solution time is of concern two numerical results are presented which demonstrate that the ammm is a useful method to analyze scattering from perfectly conducting plates

application of adaptive basis functions for a diagonal moment matrix solution of arbitrarily shaped three dimensional conducting body problems we present the application of specially constructed adaptive basis functions that generate a diagonal matrix in the method of moments solution procedure for the calculation of scattered electromagnetic fields from arbitrarily shaped conducting bodies excited by a plane electromagnetic wave the arbitrary body is modeled using planar triangular patches the crucial step in the solution procedure is the construction of the adaptive basis functions to generate the diagonal matrix this task is accomplished with the help of well known rwg basis functions the solution thus obtained is very efficient accurate and applicable to truly arbitrary bodies several numerical examples are presented to validate the new method

exceptional points of an endomorphism of the projective plane let f an endomorphism of of degree 1 we show how to obtain a bound depending only on n on the number of codimension two subspaces in which are completely invariant for f where l is completely invariant for f means that f 1 l l set theoretically

dynamics of bimeromorphic maps of surfaces we classify bimeromorphic self maps f x inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i of compact kahler surfaces x in terms of their actions f h 1 1 x inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i on cohomology we observe that the growth rate of f n is invariant under bimeromorphic conjugacy and that by conjugating one can always arrange that f n f n we show that the sequence f n can be bounded grow linearly grow quadratically or grow exponentially in the first three cases we show that after conjugating f is an automorphism virtually isotopic to the identity f preserves a rational fibration or f preserves an elliptic fibration respectively in the last case we show that there is a unique up to scaling expanding eigenvector for f that is nef and that f is bimeromorphically conjugate to an automorphism if and only if 2 0 we go on in this case to construct a dynamically natural positive current representing and we study the growth rate of periodic orbits of f we conclude by illustrating our results with a particular family of examples

sur les groupes de transformations birationnelles des surfaces ni agissant par transformations bi rationnelles sur les surfaces complexes compactes kahl eriennes nous mon trons a que le groupe des transformations birationnelles d une surface satisfait l alternative de tits b que les actions birationnelles de groupes de kazhdan sur les surfaces sont toutes birationnellement conjugu eesa des actions homographiques sur le plan projectif et c que si f et g sont deux transformations birationnelles de surfaces qui commutent alors ou bien f pr eserve un pinceau de courbes ou bien l un des it er es g m de g m 0 concide avec un it er e f n de f n 2 z let s be any compact complex kahler surface and bir s the group of birational transformations of s we study the structure of nitely genera ted subgroups of bir s and prove three main results a bir s satises the tits alternative b if a group with kazhdan property t acts on a compact kahler surface by birational transformations then the action is conjugate to an action by linear projective transformations on the projec tive plane and c if f is an element of bir s which does not preserve any pencil of curves and if g commutes with f then a positive iterate of g coincides with an iterate of f the second statement provides a positive answer to zimmer s conjecture for birational actions on surfaces

comments on application of fft and the conjugate gradient method for the solution of electromagnetic radiation from electrically large and small conducting bodies in the original paper ibid vol ap 34 no 5 p 635 40 1986 t k sarkar et al describe how the fast fourier transform fft and the conjugate gradient method cgm can be used to efficiently make use of the convolutional form of the electric field integral equation for straight wire antennas the commentators agree that the cgm converges more rapidly than the previously used method of steepest descent and the spectral iteration technique they also point out that experience with the dyadic green s function encountered in the two dimensional transverse electric te and three dimensional equations for dielectric bodies is not so encouraging it is found that for these cases the cgm convergence can be quite slow degrading the efficiency of the algorithm in addition the solution obtained using the pulse basis point matching method to discretize the integral equation are very inaccurate for the two dimensional te and three dimensional cases a reply to these comments from the authors of the original paper is included

improved impedance matrix localization method em problems moment method calculations have the well known limitations of requiring excessive storage and execution times for even modestly large electromagnetics problems the impedance matrix localization iml method was introduced as a modification to standard moment method calculations to ease these limitations it utilizes a matrix transformation which effectively changes the basis testing functions into ones resembling traveling waves an improved method that uses an orthogonal transformation to generate standing wave like basis functions is presented here remarkable improvements are achieved in the numerical stability of the method and in its compatibility with iterative solvers furthermore the correspondence of the large elements in this matrix to geometrical theory of diffraction gtd terms is strengthened as is the possibility of further increasing the speed of iterative solutions by constructing preconditioners based on the pattern of nonzero matrix elements

application of wavelets on the interval to the analysis of thin wire antennas and scatterers an analysis of thin wire antennas and scatterers using orthogonal wavelets on interval 0 1 is presented the thin wire version of the electric field integral equation efle is solved by the hybrid wavelet expansion and boundary element method hwbm maps between the curved solution domains and the interval 0 1 are established by using the geometrical representation of the boundary element method bem by virtue of these maps bases over the curved solution domains are derived from orthogonal wavelets on 0 1 that are used to expand the unknown current over the wires the utilization of the wavelets on 0 1 circumvents the difficulties in the application of the wavelets on the real line to finite domain problems and has no periodicity constraint to the unknown function that is usually imposed by the periodic wavelets numerical examples are provided for a variety of thin wire antennas and scatterers

integral points and relative sizes of coordinates of orbits in mathbb p n we give a generalization to higher dimensions of silverman s result on finiteness of integer points in orbits assuming vojta s conjecture we prove a sufficient condition for morphisms on mathbb p n so that s d integral points in each orbit are zariski non dense this condition is geometric and for dimension 1 it corresponds precisely to silverman s hypothesis that the second iterate of the map is not a polynomial in fact we will prove a more precise formulation comparing local heights outside s to the global height for hyperplanes this amounts to comparing logarithmic sizes of the coordinates generalizing silverman s precise version in dimension 1 we also discuss a variant where we can conclude that integral points in orbits are finite rather than just zariski non dense further we show unconditional results and examples using schmidt s subspace theorem and known cases of lang vojta conjecture we end with some extensions to the case of rational maps and to the case when the arithmetic of the orbit under one map is controlled by the geometric properties of another we include many explicit examples to illustrate different behaviors of integral points in orbits in higher dimensions

rapid solution of integral equations of scattering theory in two dimensions abstract the present paper describes an algorithm for rapid solution of boundary value problems for the helmholtz equation in two dimensions based on iteratively solving integral equations of scattering theory cpu time requirements of previously published algorithms of this type are of the order n 2 where n is the number of nodes in the discretization of the boundary of the scatterer the cpu time requirements of the algorithm of the present paper are n 4 3 and can be further reduced making it considerably more practical for large scale problems

rational surface automorphisms preserving cuspidal anticanonical curves this article is concerned with automorphisms on rational surfaces we develop a method for constructing automorphisms in terms of the concept of realization of orbit data and show that any automorphism preserving a cuspidal anticanonical curve is constructed from a realization of orbit data moreover some properties of automorphisms are discussed

linearity of the exceptional set for maps of p k c let f be a holomorphic self map of p k c with degree larger than 2 we show that its exceptional set is a finite union of linear subspaces

real arnold complexity versus real topological entropy for birational transformations we consider a family of birational measure preserving transformations of two variables depending on one parameter for which simple rational expressions with integer coefficients for the exact expression of the dynamical zeta function have been conjectured together with an equality between the multiplicative rate of growth of the arnold complexity and the exponential of the topological entropy this identification takes place for the birational mapping seen as a mapping bearing on two complex variables acting in a complex projective space we revisit this identification between these two quite universal complexities by considering now the mapping as a mapping bearing on two real variables the definitions of the two previous topological complexities arnold complexity and topological entropy are modified according to this real variables point of view most of the universality is lost however the results presented here are again in agreement with an identification between the multiplicative rate of growth of some real arnold complexity and the exponential of some real topological entropy a detailed analysis of this real arnold complexity as a function of the parameter of this family of birational transformations of two variables is given one can also slightly modify the definition of the dynamical zeta function introducing a real dynamical zeta function associated with the counting of the real cycles only similarly one can also introduce some real arnold complexity generating functions we show that several of these two real generating functions seem to have the same singularities furthermore we actually conjecture several simple rational expressions for them yielding again algebraic values for the exponential of the real topological entropy in particular when the parameter of our family of birational transformations becomes large we obtain two interesting compatible nontrivial rational expressions these rational results for real mappings cannot be understood in terms of any obvious markov s partition or symbolic dynamics hyperbolic systems interpretation the birational transformation is far from being hyperbolic it is measure preserving it would be useful to know whether this relation between the arnold complexity and the topological entropy as well as the rationality of the degree generating functions and dynamical zeta functions are a consequence of the measure preserving property of the mapping

periodic points for area preserving birational maps of surfaces it is a basic problem to count the number of periodic points of a surface mapping since the growth rate of this number as the period tends to infinity is an important dynamical invariant however this problem becomes difficult when the map admits curves of periodic points in this situation we give a precise estimate of the number of isolated periodic points for an area preserving birational map of a projective complex surface

potts models on hierarchical lattices and renormalization group dynamics we prove that the generator of the renormalization group of potts models on hierarchical lattices can be represented by a rational map acting on a finite dimensional product of complex projective spaces in this framework we can also consider models with an applied external magnetic field and multiple spin interactions we use recent results regarding iteration of rational maps in several complex variables to show that for some class of hierarchical lattices lee yang and fisher zeros belong to the unstable set of the renormalization map

a new mass for asymptotically flat manifolds abstract in this paper we introduce a mass for asymptotically flat manifolds by using the gauss bonnet curvature we first prove that the mass is well defined and is a geometric invariant if the gauss bonnet curvature is integrable and the decay order satisfies n 4 3 motivated by an elegant idea of lam 35 we then show a positive mass theorem for this new mass for asymptotically flat graphs over r n moreover we obtain various penrose type inequalities in this case in these penrose type inequalities we establish relationship between the new mass and various geometric integrals

conformal modulus and its reduction via mean curvatures abstract this note shows that the conformal modulus of a condenser and the reduced conformal modulus of a compact subset in r n 2 are naturally but optimally tied with the corresponding graphic mass and the semi diameter through mean curvatures

proof of the positive mass theorem ii the positive mass theorem states that for a nontrivial isolated physical system the total energy which includes contributions from both matter and gravitation is positive this assertion was demonstrated in our previous paper in the important case when the space time admits a maximal slice here this assumption is removed and the general theorem is demonstrated abstracts of the results of this paper appeared in 11 and 13

real and complex dynamics of a family of birational maps of the plane the golden mean subshift we describe the real dynamics of a family of birational mappings of the plane by combining complex intersection theory and techniques from smooth dynamical systems we are able to give an essentially complete account of the behavior of both wandering and nonwandering orbits in particular the golden mean subshift provides a topological model for the dynamics on the nonwandering set while the mappings are not hyperbolic they are shown to possess many of the structures associated with hyperbolicity

optimal grouping of basis functions when using the method of moments to solve scattering problems a crucial factor is the appropriate choice of basis functions a judicious choice of the basis functions can lead to a sparse impedance matrix or to a sparse solution vector both of which will reduce the computational burden however pulse basis functions are usually used both due to their simple description and because there are already many software tools written for this aim therefore some efforts have been invested in transforming a formulation that is the result of using pulse basis functions to one using preferred basis functions this transform produces grouping of the pulse functions into a new more appropriate set of basis functions this paper discusses the search for an optimal grouping where optimality is measured in terms of the sparseness of the resulting solution vector various options are demonstrated and discussed and numerical examples are given it is important to note that the optimal grouping suggested herein is expected to be optimal on the average when tried on many sample problems adjusting the optimization to a more specific set of problems and scenarios will lead to even better performances and is a viable option in practical cases

rational dynamical zeta functions for birational transformations we propose a conjecture for the exact expression of the unweighted dynamical zeta function for a family of birational transformations of two variables depending on two parameters this conjectured function is a simple rational expression with integer coecients this yields an algebraic value for the topological entropy furthermore the generating function for the arnold complexity is also conjectured to be a rational expression with integer coecients with the same singularities as for the dynamical zeta function this leads at least in this example to an equality between the asymptotic of the arnold complexity and the exponential of the topological entropy we also give a semi numerical method to eectively compute the arnold complexity c 1999 published by elsevier science b v all rights reserved

the gbc mass for asymptotically hyperbolic manifolds the paper consists of two parts in the first part by using the gauss bonnet curvature which is a natural generalization of the scalar curvature we introduce a higher order mass the gauss bonnet chern mass m mathbb h k for asymptotically hyperbolic manifolds and show that it is a geometric invariant moreover we prove a positive mass theorem for this new mass for asymptotically hyperbolic graphs and establish a relationship between the corresponding penrose type inequality for this mass and weighted alexandrov fenchel inequalities in the hyperbolic space mathbb h n in the second part we establish these weighted alexandrov fenchel inequalities in mathbb h n for any horospherical convex hypersurface sigma begin aligned int sigma v sigma 2k 1 d mu ge c n 1 2k 1 omega n 1 left left frac sigma omega n 1 right frac n k n 1 left frac sigma omega n 1 right frac n 2k k n 1 right k end aligned where sigma j is the j th mean curvature of sigma subset mathbb h n v cosh r and sigma is the area of sigma as an application we obtain an optimal penrose type inequality for the new mass defined in the first part begin aligned m k mathbb h ge frac 1 2 k left left frac sigma omega n 1 right frac n k n 1 left frac sigma omega n 1 right frac n 2k k n 1 right k end aligned for asymptotically hyperbolic graphs with a horizon type boundary sigma provided that a dominant energy type condition tilde l k ge 0 holds both inequalities are optimal

impedance matrix compression using adaptively constructed basis functions wavelet expansions have been employed recently in numerical solutions of commonly used frequency domain integral equations in this paper we propose a novel method for integrating wavelet based transforms into existing numerical solvers the newly proposed method differs from the presently used ones in two ways first the transformation is affected by means of a digital filtering approach this approach renders the transform algorithm adaptive and facilitates the derivation of a basis which best suits the problem at hand second the conventional thresholding procedure applied to the impedance matrix is substituted for by a compression process in which only the significant terms in the expansion of the yet unknown current are retained and subsequently derived numerical results for a few tm scattering problems are included to demonstrate the advantages of the proposed method over the presently used ones

the equality case of the penrose inequality for asymptotically flat graphs we prove the equality case of the penrose inequality in all dimensions for asymptotically flat hypersurfaces it was re cently proven by g lam 19 that the penrose inequality holds for asymptotically flat graphical hypersurfaces in euclidean space with non negative scalar curvature and with a minimal boundary our main theorem states that if the equality holds then the hy persurface is a schwarzschild solution as part of our proof we show that asymptotically flat graphical hypersurfaces with a min imal boundary and non negative scalar curvature must be mean convex using the argument that we developed in 15 this en ables us to obtain the ellipticity for the linearized scalar curvature operator and to establish the strong maximum principles for the scalar curvature equation

analysis of linear arrays using the adaptive basis functions diagonal moment matrix technique an efficient algorithm is proposed for the analysis of large finite arrays using the adaptive basis functions diagonal moment matrix technique adaptive basis functions constructed using clusters spanning over an array element are used to generate a highly diagonally dominant moment matrix the physical interpretation of the constructed set of adaptive basis functions is discussed the new matrix equation is solved iteratively in a way that only the significant mutual impedances are considered the proposed algorithm is applied to a linear array of bow tie antennas and results are compared to those obtained using the direct moment method solution and exhibit very good agreement the relative computational time improves as the array size increases compared to the conventional moment method solution a speedup factor of more than 100 is achieved for an array of 32 elements

factorization properties of birational mappings abstract we analyse birational mappings generated by transformations on q q matrices which correspond respectively to two kinds of transformations the matrix inversion and a permutation of the entries of the q q matrix remarkable factorization properties emerge for quite general involutive permutations it is shown that factorization properties do exist even for birational transformations associated with noninvolutive permutations of entries of q q matrices and even for more general transformation which are rational transformations but no longer birational the existence of factorization relations independent of q the size of the matrices is underlined the relations between the polynomial growth of the complexity of the iterations the existence of recursions in a single variable and the integrability of the mappings are sketched for the permutations yielding these properties all these results show that permutations of the entries of the matrix yielding factorization properties are not so rare in contrast the occurrence of recursions in a single variable or of the polynomial growth of the complexity are of course less frequent but not completely exceptional

adaptive multiscale moment method ammm for analysis of scattering from three dimensional perfectly conducting structures the adaptive multiscale moment method ammm is presented for the analysis of scattering from three dimensional 3d perfectly conducting bodies this algorithm employs the conventional moment method mm using the subsectional triangular patch basis functions and a special matrix transformation which is derived from solving the fredholm equation of the first kind by the multiscale technique this methodology is more suitable for problems where the matrix solution time is much greater than the matrix fill time the widely used triangular patch vector basis functions developed by rao et al 1982 are used for expansion and testing functions in the conventional mm the objective here is to compress the unknowns in existing mm codes which solves for the currents crossing the edges of the triangular patch basis functions by use of a matrix transformation the currents source terms and impedance matrix can be arranged in the form of different scales from one scale to another scale the initial guess for the solution can be predicted according to the properties of the multiscale technique ammm can reduce automatically the size of the linear equations so as to improve the efficiency of the conventional mm the basic difference between this methodology and the other wavelet based techniques that have been presented so far is that we apply the compression not to the impedance matrix but to the solution itself directly in an iterative fashion even though it is an unknown two numerical results are presented which demonstrate that the ammm is a useful method for analysis of electromagnetic scattering from arbitrary shaped 3d perfectly conducting bodies

growth complexity spectrum of some discrete dynamical systems we first study the iteration of birational mappings generated by the composition of the matrix inversion and of a permutation of the entries of 3 3 matrices and consider the degree d n of the numerators or denominators of the corresponding successive rational expressions for the nth iterate the growth of this degree is generically exponential with n d n n is called the growth complexity we introduce a semi numerical analysis which enables to compute these growth complexities for all the 9 possible birational transformations these growth complexities correspond to a spectrum of eighteen algebraic values we then drastically generalize these results replacing permutations of the entries by homogeneous polynomial transformations of the entries possibly depending on many parameters again it is shown that the associated birational or even rational transformations yield algebraic values for their growth complexities 1999 elsevier science b v all rights reserved

the wavelet transform time frequency localization and signal analysis two different procedures for effecting a frequency analysis of a time dependent signal locally in time are studied the first procedure is the short time or windowed fourier transform the second is the wavelet transform in which high frequency components are studied with sharper time resolution than low frequency components the similarities and the differences between these two methods are discussed for both schemes a detailed study is made of the reconstruction method and its stability as a function of the chosen time frequency density finally the notion of time frequency localization is made precise within this framework by two localization theorems

on the use of the singular value decomposition in the fast multipole method the authors present a new matrix compression algorithm to improve the efficiency of the fast multipole method fmm the method is based on the application of the singular value decomposition svd to the plane wave fmm aggregation matrices these matrices are low ranked which is exploited to provide alternative sets of orthonormal singular basis functions obtained as linear combinations of the original basis by choosing only the most relevant singular functions a much more compact representation is obtained to accurately handle the interactions between the fmm groups the new formulation provides a reduction close to one order of magnitude both in computational cost and memory requirements with a moderate impact on the accuracy of the solution

an accelerated dft mom for the analysis of large finite periodic antenna arrays a novel approach which utilizes a discrete fourier transform dft based preconditioner in an iterative moment method mom solution of the governing integral equation for the electromagnetic em problems of the radiation and scattering from large finite periodic antenna arrays is presented the preconditioner is primarily constructed via the block diagonal approximation of the transformed impedance matrix in the dft domain in which that matrix becomes sparse and nearly diagonal due to the properties of the dft basis functions the preconditioner in the original spatial domain is simply obtained by the inverse dft this preconditioner is shown to significantly improve the convergence rate of the iterative solution moreover it is rather simple to implement and requires relatively small extra computational cost the performance of the present approach is illustrated through numerical results

periodic points of birational transformations on projective surfaces we give a classification of birational transformations on smooth projective surfaces which have a zariski dense set of noncritical periodic points in particular we show that if the first dynamical degree is greater than one the union of all noncritical periodic orbits is zariski dense

numerical convergence in periodic method of moments analysis of frequency selective surfaces based on wire elements we present a subdomain formulation of the periodic method of moments pmm with thin wire kernel for analyzing frequency selective surfaces fsss with rectilinear wire type elements analysis of the convergence of the impedance matrix for a fss with aligned unidirectional elements indicates the effect of individual oscillatory and decaying components for the individual impedance elements of this fss we prove and demonstrate the universality of their envelopes as a function of shell size in the spectral domain for n wire segments the pmm converges according to o n sup 4 the dependence on the order of polynomial basis functions shows a geometric progression the theory is also applied to a single layer fss having asymmetrically split segmented rings

energy and invariant measures for birational surface maps given a birational self map of a compact complex surface it is useful to find an invariant measure that relates the dynamics of the map to its action on cohomology under a very weak hypothesis on the map we show how to construct such a measure the main point in the construction is to make sense of the wedge product of two positive closed 1 1 currents we are able to do this in our case because local potentials for each current have finite energy with respect to the other our methods also suffice to show that the resulting measure is mixing does not charge curves and has nonzero lyapunov exponents

infinitesimal deformations of rational surface automorphisms if x is a rational surface without nonzero holomorphic vector field and f is an automorphism of x we study in several examples the zariski tangent space of the local deformation space of the pair x f

invariant curves for birational surface maps we classify invariant curves for birational surface maps that are expanding on cohomology when the expansion is exponential the arithmetic genus of an invariant curve is at most one this implies severe constraints on both the type and number of irreducible components of the curve in the case of an invariant curve with genus equal to one we show that there is an associated invariant meromorphic two form

real polynomial diffeomorphisms with maximal entropy tangencies like the henon diffeomorphisms of r2 the quadratic maps of r have also played a central role in the field of dynamical systems these two families of dynamical systems fit together naturally which raises the question of the extent to which the dynamics is similar one difference is that our knowledge of these quadratic maps is much more thorough than our knowledge of these quadratic diffeomorphisms substantial progress in the theory of quadratic maps has led to a rather complete theoretical picture of their dynamics and an understanding of how the dynamics varies with the parameter despite significant recent progress in the theory of henon diffeomorphisms due to benedicks and carleson and many others there are still many phenomena that are not nearly so well understood in this two dimensional setting as they are for quadratic maps one phenomenon which illustrates the difference in the extent of our knowledge in dimensions one and two is the dependence of the complexity of the system on parameters in one dimension the nature of this dependence is understood and the answer is summarized by the principle of monotonicity loosely stated this is the assertion that the complexity of fa does not

real topological entropy versus metric entropy for birational measure preserving transformations abstract we consider a family of birational measure preserving transformations of two complex variables depending on one parameter for which simple rational expressions for the dynamical zeta function have been conjectured together with an equality between the topological entropy and the logarithm of the arnold complexity divided by the number of iterations similar results have been obtained for the adaptation of these two concepts to dynamical systems of real variables yielding to introduce a real topological entropy and a real arnold complexity we try to compare here the kolmogorov sinai metric entropy and this real arnold complexity or real topological entropy on this particular example of a one parameter dependent birational transformation of two variables more precisely we analyze using an infinite precision calculation the lyapunov characteristic exponents for various values of the parameter of the birational transformation in order to compare these results with the ones for the real arnold complexity we find a quite surprising result for this very birational example and in fact for a large set of birational measure preserving mappings generated by involutions the lyapunov characteristic exponents seem to be equal to zero or at least extremely small for all the orbits we have considered and for all values of the parameter birational measure preserving transformations generated by involutions could thus allow to better understand the difference between the topological description and the probabilistic description of discrete dynamical systems many birational measure preserving transformations generated by involutions seem to provide examples of discrete dynamical systems which can be topologically chaotic while they are metrically almost quasi periodic heuristically this can be understood as a consequence of the fact that their orbits seem to form some kind of transcendental foliation of the two dimensional space of variables

a birational mapping with a strange attractor post critical set and covariant curves we consider some two dimensional birational transformations one of them is a birational deformation of the henon map for some of these birational mappings the post critical set i e the iterates of the critical set is infinite and we show that this gives straightforwardly the algebraic covariant curves of the transformation when they exist these covariant curves are used to build the preserved meromorphic 2 form one may also have an infinite post critical set yielding a covariant curve which is not algebraic transcendental for two of the birational mappings considered the post critical set is finite and we claim that there is no algebraic covariant curve and no preserved meromorphic 2 form for these two mappings with finite post critical sets attracting sets occur and we show that they pass the usual tests lyapunov exponents and the fractal dimension for being strange attractors the strange attractor of one of these two mappings is unbounded