maximal abelian subalgebras of e p q algebras abstract maximal abclian subalgebras masas of one of the classical real inhomogencous lie algebras are constructed namely those of the pseudoeuclidean lie algebra e p q use is made of the semidireet sum structure of e p q with the translations t p q as an abclian ideal we first construct splitting masas that arc themselves direct sums of abelian subalgebras of o p q and of subalgebras of t p q the splitting subalgebras are used to construct the complementary nonsplitting ones here the results are less complete than in the splitting case we present general decomposition theorems and construct indecomposable masas for all algebras e p q p q 0 the case of q 0 and 1 were treated earlier in a physical context the case q 2 is analyzed here in detail as an illustration of the general results

area and hausdorff dimension of julia sets of entire functions we show the julia set of a sin z has positive area and the action of a sin z on its julia set is not ergodic the julia set of a exp z has hausdorff dimension two but in the presence of an attracting periodic cycle its area is zero

contractions of lie algebras and separation of variables the wigner inonu contraction from the rotation group o 3 to the euclidean group e 2 is used to relate the separation of variables in the laplace beltrami operators on two corresponding homogeneous spaces different realizations of the contraction take the two separable coordinate systems on the sphere to the four on the plane

herman s condition and siegel disks of bi critical polynomials we extend a theorem of herman from the case of unicritical polynomials to the case of polynomials with two finite critical values this theorem states that siegel disks of such polynomials under a diophantine condition called herman s condition on the rotation number must have a critical point on their boundaries

dynamics of meromorphic functions with direct or logarithmic singularities let f be a transcendental meromorphic function and denote by j f the julia set and by i f the escaping set we show that if f has a direct singularity over infinity then i f has an unbounded component and i f j f contains continua moreover under this hypothesis i f j f has an unbounded component if and only if f has no baker wandering domain if f has a logarithmic singularity over infinity then the upper box dimension of i f j f is 2 and the hausdorff dimension of j f is strictly greater than 1 the above theorems are deduced from more general results concerning functions which have direct or logarithmic tracts but which need not be meromorphic in the plane these results are obtained by using a generalization of wiman valiron theory this method is also applied to complex differential equations

coherent states for the hydrogen atom discrete and continuous spectra we construct the systems of generalized coherent states for the discrete and continuous spectra of the hydrogen atom these systems are expressed in elementary functions and are invariant under the so 3 2 discrete spectrum and so 4 1 continuous spectrum subgroups of the dynamical symmetry group so 4 2 of the hydrogen atom both systems of coherent states are particular cases of the kernel of integral operator which intertwines irreducible representations of the so 4 2 group

slow escaping points of meromorphic functions we show that for any transcendental meromorphic function f there is a point z in the julia set of f such that the iterates f n z escape that is tend to arbitrarily slowly the proof uses new covering results for analytic functions we also introduce several slow escaping sets in each of which f n z tends to at a bounded rate and establish the connections between these sets and the julia set of f to do this we show that the iterates of f satisfy a strong distortion estimate in all types of escaping fatou components except one which we call a quasi nested wandering domain we give examples to show how varied the structures of these slow escaping sets can be

critical points on the boundaries of siegel disks let be a polynomial map of the riemann sphere of degree at least two we prove that if has a siegel disk g on which the rotation number satisfies a diophantine condition then the boundary of g contains a critical point

existence of simply connected algebraic surfaces of general type with positive and zero indices abstract in the classification problem of algebraic surfaces of general type an important conjecture states that for simply connected such surfaces chern numbers satisfy the inequality c12 2c2 or equivalently the index 0 we disprove this conjecture by computing fundamental groups of galois coverings corresponding to generic cp2 projections of projective embeddings of cp1 cp1 related to linear systems unk al1 bl2 unk a 3 b 2 also we proved the existence of simply connected minimal surfaces of general type with zero index e g c12 2c2 previously it was conjectured that these are exactly the surfaces uniformizable in the polydisk so this conjecture is also disproved

maximal abelian subgroups of the isometry and conformal groups of euclidean and minkowski spaces the maximal abelian subalgebras masas of the euclidean and pseudo euclidean lie algebras are classified into conjugacy classes under the action of the corresponding lie groups and and also under the conformal groups and respectively the results are presented in terms of decomposition theorems for orthogonally indecomposable masas exist only for p 1 and p 2 for on the other hand orthogonally indecomposable masas exist for all values of p the results are used to construct new coordinate systems in which wave equations and hamilton jacobi equations allow the separation of variables

absence of wandering domains for some real entire functions with bounded singular sets let f be a real entire function whose set s f of singular values is real and bounded we show that if f satisfies a certain function theoretic condition the sector condition then f has no wandering domains our result includes all maps of the form z mapsto lambda frac sinh z z a with lambda 0 and a in mathbb r we also show the absence of wandering domains for certain non real entire functions for which s f is bounded and f n s f rightarrow infty uniformly as a special case of our theorem we give a short elementary and non technical proof that the julia set of the exponential map f z e z is the entire complex plane furthermore we apply similar methods to extend a result of bergweiler concerning baker domains of entire functions and their relation to the postsingular set to the case of meromorphic functions

a bound on the number of rationally invisible repelling orbits abstract we consider entire transcendental maps with bounded set of singular values such that periodic rays exist and land for such maps we prove a refined version of the fatou shishikura inequality which takes into account rationally invisible periodic orbits that is repelling cycles which are not landing points of any periodic ray more precisely if there are q singular orbits then the sum of the number of attracting parabolic siegel cremer or rationally invisible orbits is bounded above by q in particular there are at most q rationally invisible repelling periodic orbits the techniques presented here also apply to the more general setting in which the function is allowed to have infinitely many singular values

the minimal dimension of maximal commutative subalgebras of full matrix algebras abstract let f be a field and a a maximal commutative subalgebra of the full matrix algebra mn f it is shown that dim a 2n 2 3 1 it is also shown that if the radical of a has cube zero then dim a 3n 2 3 4 and that this result is best possible for infinitely many natural numbers n

separation of variables and subgroup bases on n dimensional hyperboloids a graphical formalism is introduced for describing subgroup type coordinates on n dimensional lorentzian hyperboloids imbedded into n 1 dimensional minkowski spaces the o n 1 group element is parametrized according to different subgroup chains involving lorentz rotation and euclidean subgroups the coordinates are then induced by the corresponding group action eigenfunctions of the laplace beltrami operator are obtained as products of jacobi functions associated legendre functions and modified bessel functions

nonlinear action of lie groups and superposition principles for nonlinear differential equations abstract recently obtained results on superposition principles for systems of ordinary nonlinear equations are reviewed in particular the general solution of the matrix riccati equation is expressed in terms of five particular solutions the study of superposition laws is related to the problem of backlund transformations for nonlinear partial differential equations to the problem of transforming nonlinear equations into linear ones and to the classification of primitive filtered lie algebras

superposition principles for matrix riccati equations a superposition rule is obtained for the matrix riccati equation mre w a wb cw wdw where w t a t b t c t and d t are real n n matrices expressing the general solution in terms of five known solutions for all n 2 the symplectic mre w wt a at d dt c bt is treated separately and a superposition rule is derived involving only four known solutions for the unitary and gl n r subcases with d a and c bt or d a and c bt respectively superposition rules are obtained involving only two solutions the derivation of these results is based upon an interpretation of the mre in terms of the action of the groups sl 2n r sp 2n r u n and gl n r on the grassman manifold gn r2n

dynamical invariance algebra of the hartmann potential the accidental degeneracy occurring in the quantum mechanical treatment of the ring shaped potential v eta sigma 2r 1 1 2q eta 2 sigma 2 r sin theta 2 is explained by constructing an su 2 dynamical invariance algebra the schrodinger equation is solved in parabolic coordinates written in the framework of the kustaanheimo stiefel transformation and the hamilton jacobi equations are solved in ordinary parabolic coordinates all finite trajectories are found to be periodic

diophantine conditions imply critical points on the boundaries of siegel disks of polynomials let f be a polynomial map of the riemann sphere of degree at least two we prove that if f has a siegel disk g on which the rotation number satisfies a diophantine condition then either the boundary b of g contains a critical point or b is a lakes of wada indecomposable continuum with one of the lakes containing a critical point consequently if the boundary b of g has only 2 complementary domains then b contains a critical point we also show without any assumption on the rotation number that each proper nondegenerate subcontinuum of the boundary b of g is tree like and any other bounded complementary domain of b is a preperiodic component of the grand orbit of g finally we establish some conditions under which b contains no periodic point

maximal abelian subalgebras of pseudounitary lie algebras abstract the task of classifying and constructing all maximal abelian subalgebras of su p q p q 1 is reduced to that of classifying orthogonally indecomposable masas these are either maximal abelian nilpotent subalgebras represented by nilpotent matrices in any finite dimensional representation or for p q they can be nonorthogonally decomposable and their study can be reduced to a construction of manss of sl p c two types of manss of su p q are shown to exist one rowed and non one rowed numerous classification theorems are proven and applied to obtain all masas of su p 1 su p 2 and su p q with p q 6 physical applications are discussed

entire functions with julia sets of positive measure let f be a transcendental entire function for which the set of critical and asymptotic values is bounded the denjoy carleman ahlfors theorem implies that if the set of all z for which f z r has n components for some r 0 then the order of f is at least n 2 more precisely we have log log m r f n 2 log r o 1 where m r f denotes the maximum modulus of f we show that if f does not grow much faster than this then the escaping set and the julia set of f have positive lebesgue measure however as soon as the order of f exceeds n 2 this need not be true the proof requires a sharpened form of an estimate of carleman and tsuji related to the denjoy carleman ahlfors theorem

third order superintegrable systems with potentials satisfying only nonlinear equations the conditions for superintegrable systems in two dimensional euclidean space admitting separation of variables in an orthogonal coordinate system and a functionally independent third order integral are studied it is shown that only systems that separate in subgroup type coordinates cartesian or polar admit potentials that can be described in terms of nonlinear special functions systems separating in parabolic or elliptic coordinates are shown to have potentials with only non movable singularities

trees and hairs for some hyperbolic entire maps of finite order let f be an entire transcendental map of finite order such that all the singularities of f 1 are contained in a compact subset of the immediate basin b of an attracting fixed point it is proved that there exist geometric coding trees of preimages of points from b with all branches convergent to points from hat mathbb c this implies that the riemann map onto b has radial limits everywhere moreover the julia set of f consists of disjoint curves hairs tending to infinity homeomorphic to a half line composed of points with a given symbolic itinerary and attached to the unique point accessible from b endpoint of the hair these facts generalize the corresponding results for exponential maps

the geometry of julia sets the long term analysis of dynamical systems inspired the study of the dynamics of families of mappings many of these investigations led to the study of the dynamics of mappings on cantor sets and on intervals julia sets play a critical role in the understanding of the dynamics of families of mappings in this paper we introduce another class of objects called hairy objects which share many properties with the cantor set and the interval they are topologically unique and admit only one embedding in the plane these uniqueness properties explain the regular occurrence of hairy objects in pictures of julia sets hairy objects are ubiquitous hairy arcs will be used to give a complete topological description of the julia sets of many members of the exponential family

hyperbolic entire functions and the eremenko lyubich class class mathcal b or not class mathcal b hyperbolicity plays an important role in the study of dynamical systems and is a key concept in the iteration of rational functions of one complex variable hyperbolic systems have also been considered in the study of transcendental entire functions there does not appear to be an agreed definition of the concept in this context due to complications arising from the non compactness of the phase space in this article we consider a natural definition of hyperbolicity that requires expanding properties on the preimage of a punctured neighbourhood of the isolated singularity we show that this definition is equivalent to another commonly used one a transcendental entire function is hyperbolic if and only if its postsingular set is a compact subset of the fatou set this leads us to propose that this notion should be used as the general definition of hyperbolicity in the context of entire functions and in particular that speaking about hyperbolicity makes sense only within the eremenko lyubich class mathcal b of transcendental entire functions with a bounded set of singular values we also considerably strengthen a recent characterisation of the class mathcal b by showing that functions outside of this class cannot be expanding with respect to a metric whose density decays at most polynomially in particular this implies that no transcendental entire function can be expanding with respect to the spherical metric finally we give a characterisation of an analogous class of functions analytic in a hyperbolic domain

absence of line fields and mane s theorem for nonrecurrent transcendental functions let f c c over cap be a transcendental meromorphic function suppose that the finite part p f boolean and c of the postsingular set of f is bounded that f has no recurrent critical points or wandering domains and that the degree of pre poles of f is uniformly bounded then we show that f supports no invariant line fields on its julia set we prove this by generalizing two results about rational functions to the transcendental setting a theorem of mane 1993 about the branching of iterated preimages of disks and a theorem of mcmullen 1994 regarding the absence of invariant line fields for measurably transitive functions both our theorems extend results previously obtained by graczyk kotus and swiatek 2004

rigidity of escaping dynamics for transcendental entire functions we prove an analog of bottcher s theorem for transcendental entire functions in the eremenko lyubich class mathcal b more precisely let f and g be entire functions with bounded sets of singular values and suppose that f and g belong to the same parameter space i e are quasiconformally equivalent in the sense of eremenko and lyubich then f and g are conjugate when restricted to the set of points that remain in some sufficiently small neighborhood of infinity under iteration furthermore this conjugacy extends to a quasiconformal self map of the plane

normal forms of elements of classical real and complex lie and jordan algebras elements of the classical complex and real lie and jordan algebras with involutions are classified into conjugacy classes under the action of the corresponding classical lie group normal forms of representatives of each conjugacy class are chosen so as to resemble the jordan normal forms of n n complex matrices for completeness similar results are given for gl n c gl n r and gl n h

contractions of lie algebras and the separation of variables interbase expansions lie algebra contractions from o n 1 to e n are used to obtain asymptotic limits of interbase expansions between bases corresponding to different subgroup chains for the group o n 1 the contractions lead to interbase expansions for different subgroup chains of the euclidean group e n they provide asymptotic formulae for quantities such as wigner rotation matrices clebsch gordan coefficients and racah coefficients

bifurcations of circle maps arnol d tongues bistability and rotation intervals we study the bifurcations of two parameter families of circle maps that are similar tofb w x x w b 2 sin 2 x mod1 the bifurcation diagram is constructed in terms of setstr wheretr is the set of parameter values b w for whichfb w has an orbit with rotation numberr we show that the known structure whenb 1 wherefb w is no longer injective and can have an interval of rotation numbers specifically the tongues overlap in a uniform monotonic manner and forr irrational tr opens into a tongue our other theorems give information about the dynamics offb w e g bistability or aperiodicity for b w in various subsets of parameter space

completely integrable relativistic hamiltonian systems and separation of variables in hermitian hyperbolic spaces the hamilton jacobi and laplace beltrami equations on the hermitian hyperbolic space hh 2 are shown to allow the separation of variables in precisely 12 classes of coordinate systems the isometry group of this two complex dimensional riemannian space su 2 1 has four mutually nonconjugate maximal abelian subgroups these subgroups are used to construct the separable coordinates explicitly all of these subgroups are two dimensional and this leads to the fact that in each separable coordinate system two of the four variables are ignorable ones the symmetry reduction of the free hh 2 hamiltonian by a maximal abelian subgroup of su 2 1 reduces this hamiltonian to one defined on an o 2 1 hyperboloid and involving a nontrivial singular potential separation of variables on hh 2 and more generally on hh n thus provides a new method of generating nontrivial completely integrable relativistic hamiltonian systems

maximal abelian subalgebras of pseudoorthogonal lie algebras abstract the classification of maximal abelian subalgebras masas of o p q is reduced to the construction of orthogonally indecomposable oid masas five types of oid masas exist and their classification amounts to an analysis of maximal abelian nilpotent subalgebras manss of o p q su p q sl n r or sl n c theorems are provided making possible the classification of all manss of o p q and low dimensional examples are analyzed in detail

maximal abelian subalgebras of real and complex symplectic lie algebras we provide guidelines for classifying maximal abelian subalgebras masa s of the symplectic lie algebras sp 2n r and sp 2n c into conjugacy classes under the lie groups sp 2n r and sp 2n c respectively the task of classifying all masa s is reduced to the classification of orthogonally indecomposable oid masa s two types of orthogonally indecomposable masa s of sp 2n c exist 1 indecomposable maximal abelian nilpotent subalgebras mans s 2 decomposable masa s their classification reduces to a classification of mans s of sl n c four types of orthogonally indecomposable masa s of sp 2n r exist 1 absolutely indecomposable masa s mans s 2 relatively indecomposable masa s their classification reduces to a classification of mans s of su p q for p q n 3 decomposable absolutely oid masa s involving mans s of sl n r 4 decomposable relatively oid masa s involving mans s of sl n 2 c for n even low dimensional cases of sp 2n f n 1 2 3 f r or c are treated exhau

the space of composants of an indecomposable continuum abstract the family of all composants of an indecomposable continuum is studied we investigate the equivalence relation induced on an indecomposable continuum by its partition into composants we show that up to borel bireducibility such an equivalence relation can be of only two types e 0 and e 1 the e 0 type being simple and the e 1 type being complicated as a consequence of this we show that each indecomposable continuum carries a borel probability measure which assigns 0 to each composant and 0 or 1 to each borel set which is the union of a family of composants in particular it follows that there is no borel transversal for the family of all composants this solves an old problem in the theory of continua we prove that all hereditarily indecomposable continua are of the complicated type that is they fall into the e 1 type we analyze the properties of being of type e 1 or of type e 0 we show using effective descriptive set theory that the first of these properties is analytic and so the second one is coanalytic we construct examples of continua of both types in fact we produce a family of indecomposable continua and use it to prove that these properties are complete analytic and complete coanalytic respectively hence non borel so they do not admit simple topological characterizations we also use continua from this family to show that an indecomposable continuum may be of type e 1 only because of the behavior of composants on a small subset of the continuum this in particular shows that certain natural approaches to solving kuratowski s problem on generic ergodicity of the composant equivalence relation will not work we finish with some open problems

hausdorff dimension and conformal measures of feigenbaum julia sets 1 1 statement of the results one of the first questions usually asked about a fractal subset of r is whether it has the maximal possible hausdorff dimension n it certainly happens if the set has positive lebesgue measure on the other hand it is easy to construct fractal sets of zero measure but of dimension n moreover this phenomenon is often observable for fractal sets produced by conformal dynamical systems iterated rational functions or kleinian groups in particular the analogy with kleinian groups suggested that the julia sets of feigenbaum maps should have hausdorff dimension two in this paper we will show that this is not always the case a quadratic polynomial or more generally a quadratic like map is called feigenbaum if it is infinitely renormalizable of bounded combinatorial type with a priori bounds see 2 for a precise definition feigenbaum polynomials are remarkable dynamical systems whose geometry has been a focus of research for the past 25 years by analogy with kleinian groups it was anticipated that their julia sets have dimension 2 and this problem has been around for a while however in this paper we will show that this is not always the case

solvable lie algebras with abelian nilradicals a procedure is presented for classifying solvable lie algebras with abelian nilradicals theorems on the structure of such algebras are proven and their centres are constructed many examples are analysed

rigidity for real polynomials we prove the topological or combinatorial rigidity property for real polynomials with all critical points real and nondegenerate which completes the last step in solving the density of axiom a conjecture in real one dimensional dynamics

a separation theorem for entire transcendental maps we study the distribution of periodic points for a wide class of maps namely entire transcendental functions of nite order and with bounded set of singular values or compositions thereof fix p n and assume that all dynamic rays which are invariant under f p land an interior p periodic point is a xed point of f p which is not the landing point of any periodic ray invariant under f p points belonging to attracting siegel or cremer cycles are examples of interior periodic points for functions as above we show that rays which are invariant under f p together with their landing points separate the plane into nitely many regions each containing exactly one interior p periodic point or one parabolic immediate basin invariant under f p this result generalizes the goldberg milnor separation theorem for polynomials gm and has several corollaries it follows for example that two periodic fatou components can always be separated by a pair of periodic rays landing together that there cannot be cremer points on the boundary of siegel disks that hidden components of a bounded siegel disk have to be either wandering domains or preperiodic to the siegel disk itself or that there are only nitely many non repelling cycles of any given period regardless of the number of singular values

pseudo orthogonal groups and integrable dynamical systems in two dimensions integrable systems in low dimensions constructed through the symmetry reduction method are studied using phase portrait and variable separation techniques in particular invariant quantities and explicit periodic solutions are determined widely applied models in physics are shown to appear as particular cases of the method

maximal abelian subalgebras of complex orthogonal lie algebras abstract the study of maximal abelian subalgebras masas of o n c is reduced to the study of orthogonally indecomposable masas their classification is in turn reduced to that of maximal abelian nilpotent subalgebras manss of sl n 2 c for n even and of o n c itself general results and guidelines are given making it possible to construct all manss and hence all masas for any given n representatives of all o n c conjugacy classes of masas of o n c are constructed for 2 n 7 as an application of the general techniques

density of hyperbolicity for classes of real transcendental entire functions and circle maps we prove density of hyperbolicity in spaces of i real transcendental entire functions bounded on the real line whose singular set is finite and real and ii transcendental self maps of the punctured plane which preserve the circle and whose singular set apart from zero and infinity is contained in the circle in particular we prove density of hyperbolicity in the famous arnol d family of circle maps and its generalizations and solve a number of other open problems for these functions including three conjectures by de melo salom ao and vargas we also prove density of real hyperbolicity for certain families as in i but without the boundedness condition our results apply in particular when the functions in question have only finitely many critical points and asymptotic singularities or when there are no asymptotic values and the degree of critical points is uniformly bounded

singularities of inner functions associated with hyperbolic maps abstract let f be a function in the eremenko lyubich class b and let u be an unbounded forward invariant fatou component of f we relate the number of singularities of an inner function associated to f u with the number of tracts of f in particular we show that if f lies in either of two large classes of functions in b and also has finitely many tracts then the number of singularities of an associated inner function is at most equal to the number of tracts of f our results imply that for hyperbolic functions of finite order there is an upper bound related to the order on the number of singularities of an associated inner function

an extended fatou shishikura inequality and wandering branch continua for polynomials abstract let p be a polynomial of degree d with julia set j p let n be the number of non repelling cycles of p by the famous fatou shishikura inequality n d 1 the goal of the paper is to improve this bound the new count includes wandering collections of non pre critical branch continua i e collections of continua or points q i j p all of whose images are pairwise disjoint contain no critical points and contain the limit sets of eval q i 3 external rays also we relate individual cycles which are either non repelling or repelling with no periodic rays landing to individual critical points that are recurrent in a weak sense a weak version of the inequality reads n n irr i eval q i 2 d 1 where n irr counts repelling cycles with no periodic rays landing at points in the cycle q i form a wandering collection b c of non pre critical branch continua 1 if b c is non empty and 0 otherwise

building blocks for quadratic julia sets we obtain results on the structure of the julia set of a quadratic polynomial p with an irrationally indifferent fixed point zo in the iterative dynamics of p in the cremer point case under the assumption that the julia set is a decomposable continuum we obtain a building block structure theorem for the corresponding julia set j j p there exists a nowhere dense subcontinuum b c j such that p b b b is the union of the impressions of a minimally invariant cantor set a of external rays b contains the critical point and b contains both the cremer point zo and its preimage in the siegel disk case under the assumption that no impression of an external ray contains the boundary of the siegel disk we obtain a similar result in this case b contains the boundary of the siegel disk properly if the critical point is not in the boundary and b contains no periodic points in both cases the julia set j is the closure of a skeleton s which is the increasing union of countably many copies of the building block b joined along preimages of copies of a critical continuum c containing the critical point in addition we prove that if p is any polynomial of degree d 2 with a siegel disk which contains no critical point on its boundary then the julia set j p is not locally connected we also observe that all quadratic polynomials which have an irrationally indifferent fixed point and a locally connected julia set have homeomorphic julia sets

dynamics of the complex standard family the complexification of the standard family of circle maps sin mod 2 whose parameter space contains the well known arnold tongues is given byf ei e 2 1 a holomorphic map of c with essential singularities at 0 and for real values of the parameters we study the dynamical plane of the familyf near the essential singularities we prove the existence ofhairsin the julia set an invariant set of curves organized by some symbolic dynamics whose points that are not endpoints tend exponentially fast to 0 or under iteration for 1 we prove that under certain conditions all hairs end up attached to the unit circle as we increase the parameter

integrable systems based on su p q homogeneous manifolds the general theory of the separation of variables in hamilton jacobi and laplace beltrami equations on the su p q hyperboloid is used to introduce completely integrable hamiltonian systems on o p q hyperboloids each of the q 1 different cartan subalgebras of su p q leads to a different integrable o p q potential different complete sets of integrals of motion are obtained for each of the integrable systems

subgroups of lie groups and separation of variables separable systems of coordinates for the helmholtz equation d e in pseudo riemannian spaces of dimension d have previously been characterized algebraically in terms of sets of commuting second order symmetry operators for the operator d they have also been characterized geometrically by the form that the metric ds2 gik x dxidxk can take we complement these characterizations by a group theoretical one in which the second order operators are related to continuous and discrete subgroups of g the symmetry group of d for d 3 we study all separable coordinates that can be characterized in terms of the lie algebra l of g and show that they are of eight types seven of which are related to the subgroup structure of g our method clearly generalizes to the case d 3 although each separable system corresponds to a pair of commuting symmetry operators there do exist pairs of commuting symmetries s1 s2 that are not associated with separable coordinates for subgroup related operators we show in detail just

abelian integrals and the reduction method for an integrable hamiltonian system a classical finite dimensional integrable hamiltonian system corresponding to the motion of a particle constrained to an n dimensional sphere n 0 x2 1 with the hamiltonian h 1 2 y2 u2 x2 e x2 where u and e are constants and y are the momenta conjugate to x is integrated using several different methods these are the following 1 the projection of geodesic free flow on a larger space namely the sphere s2n 1 for e 0 the flow is obtained in terms of elementary functions 2 separation of variables in the hamilton jacobi equation in elliptic coordinates or alternatively the use of a complete set of integrals of motion in involution to reduce hamilton s equations to quadratures the flow is obtained in terms of abelian integrals which are then inverted in terms of generalized functions the relation between the different methods and results is clarified using methods of algebraic geometry in particular the geometry of quadrics

dynamic rays of bounded type entire functions we construct an entire function in the eremenko lyubich class b whose julia set has only bounded path components this answers a question of eremenko from 1989 in the negative on the other hand we show that for many functions in b in particular those of nite order every escaping point can be connected to 1 by a curve of escaping points this gives a partial positive answer to the aforementioned question of eremenko and answers a question of fatou from 1926

complex analytic dynamics on the riemann sphere dichotomie dynamique de fatou et julia points periodiques consequences du theoreme de montel l ensemble de julia est la fermeture de l ensemble des points periodiques repulseurs resultats classiques sur l ensemble de fatou classification de sullivan de l ensemble de fatou une condition pour le developpement sur l ensemble de julia la dynamique des polynomes l ensemble de mandelbrot et le travail de douady et hubbard le theoreme d application de riemann mesurable et la dynamique analytique

determination of the symmetries characterising separable systems in euclidean spaces the additively separable coordinate systems for the hamilton jacobi equation of classical mechanics in n dimensional complex euclidean space cn are characterised explicitly in terms of the lie symmetry algebra of cn specifically a formula is given by which the n separation constants regarded as functions of position and momentum can be expressed as members of the enveloping algebra of cn i e as elements of the vector space spanned by products of the members of the lie algebra of cn by quantising these n integrals of the motion the author easily obtains the commuting sets of n partial differential operators characterising the multiplicatively separable systems for the corresponding helmholtz equation a package including this formula has been implemented in the symbolic language macsyma the resulting program is also capable of calculating many of the time consuming details separation equations etc of separation on cn

quantum superintegrability and exact solvability in n dimensions a family of maximally superintegrable systems containing the coulomb atom as a special case is constructed in n dimensional euclidean space two different sets of n commuting second order operators are found overlapping in the hamiltonian alone the system is separable in several coordinate systems and is shown to be exactly solvable it is solved in terms of classical orthogonal polynomials the hamiltonian and n further operators are shown to lie in the enveloping algebra of a hidden affine lie algebra

contractions of lie algebras and separation of variables the n dimensional sphere inonu wigner contractions from the rotation group o n 1 to the euclidean group e n are used to relate the separation of variables in laplace beltrami operators on n dimensional spheres and euclidean spaces respectively in this article we consider all subgroup type coordinates corresponding to different chains of subgroups of o n 1 and e n respectively in particular the contractions relate the graphical formalism of trees on spheres to the clusters on euclidean spaces introduced in this article the contractions are considered analytically on several levels the vector fields realizing the lie algebras the complete sets of commuting operators characterizing separable coordinate systems the coordinate systems themselves and the separated eigenfunctions

constructing entire functions by quasiconformal folding we give a method for constructing transcendental entire functions with good control of both the singular values of f and the geometry of f among other applications we construct a function f with bounded singular set whose fatou set contains a wandering domain