invariant metrics of positive ricci curvature on principal bundles let y be a compact connected riemannian manifold with a metric of positive ricci curvature let pi p rightarrow y be a principal bundle over y with compact connected structure group g if the fundamental group of p is finite we show that p admits a g invariant metric with positive ricci curvature so that pi is a riemannian submersion

topological obstructions to nonnegative curvature we find new obstructions to the existence of complete riemannian metric of nonnegative sectional curvature on manifolds with infinite fundamental groups in particular we construct many examples of vector bundles whose total spaces admit no nonnegatively curved metrics

a family of highly symmetric codes we define a class of codes admitting a large automorphism group this family contains the binary extended hamming code the hexacode the golay codes the pless symmetry codes as well as the 16 4 12 sub 8 codes constructed by marcugini et al a computer search resulted in the construction of codes with new parameters 28 7 18 sub 8 32 8 20 sub 8 and 39 13 17 sub 4 belonging to this family

a class of linear codes with good parameters a construction of linear codes with good parameters is given based on brouwer s table more than 100 new codes are obtained from our construction

volumes of small balls on open manifolds lower bounds and examples question under what curvature assumptions on a complete open manifold is the volume of balls of a fixed radius bounded below independent of the center point xx two theorems establish such assumptions and two examples sharply limit their weakening in particular we give an example of a metric on r4 extending to higher dimensions of positive ricci curvature whose sectional curvatures decay to 0 and such that the volume of balls goes uniformly to 0 as the center goes to infinity 0 introduction the following question came up in discussions of the first author with peter li under what curvature assumptions on a complete open manifold m can one find a constant c m r such that the volume of balls b x r can be bounded from below independent of the midpoint x 0 1 vol b x r c m r note that in the presence of a lower bound on the ricci curvature one needs to prove the above inequality only for some r since the bishop gromov volume comparison theorem gr p 124 then gives a bound c m r for r c n rn in dimension 2 theorem a below is a satisfyingpositive answer since the assumptions do not imply a lower bound on the injectivity radius while any weakening of the assumptions leads to immediate counterexamples theorem a if m2 is complete with curvature k o then there exist c m such that for radii r c m r2 in higher dimensions however we have example 1 there are convex hypersurfaces md in rd 1 such that for d 3 no lower bounds as in 0 1 exist for md in dimensions 2 we therefore assume an upper curvature bound but then one gets uniform control even on the injectivity radius the following theorem has been rediscovered several times received by the editors august 10 1987 1980 mathematics subject classification 1985 revision primary 53c20 secondary 53a07 research supported by nsf grant mcs 7g01780 and the sloan foundation both authors were supported by invitations to the i h e s and the m s r i r 1988 american mathematical society 0002 9947 88 1 00 25 per page

lifting group actions and nonnegative curvature we examine the question when a group acting by cohomogeneity one on the base of a principal g bundle can be lifted to the total space and commutes with the action by g we answer this question completely when the base of the principle bundle is 4 dimensional

a two dimensional polynomial mapping with a wandering fatou component we show that there exist polynomial endomorphisms of 2 possessing a wandering fatou component these mappings are polynomial skew products and can be chosen to extend holomorphically of p 2 c we also find real examples with wandering domains in 2 the proof is based on parabolic implosion techniques and is based on an original idea of m lyubich

fatou and julia sets for entire mappings in bbb c k the fatou set is just made of basins of attraction correspondingto attracting cycles and that hyperbolic periodic cycles are dense in the julia set similar results are obtained in fs3 fs4 for other classes of mappings seetheorems 2 9 and 2 10 below in bu buzzard proved a version of the kupka smale theorem for

quasiconformal homeomorphisms and dynamics i solution of the fatou julia problem on wandering domains on demontre un resultat dynamique sur l iteration d une application analytique complexe sur la sphere de riemann

polynomial diffeomorphisms of ii stable manifolds and recurrence f x y y y ax c the diffeomorphisms in this class have complicated dynamics in particular they have periodic points of infinitely many periods and positive topological entropy see fm s we can distinguish between these classes by considering the growth of the degrees of iterates of the diffeomorphism define deg f to be the maximum of the degrees of the coordinate functions this quantity is not a conjugacy invariant of f hence not a dynamical invariant we can construct a conjugacy invariant which we call the dynamical degree as follows

dynamics of transcendental henon maps the dynamics of transcendental functions in the complex plane has received a significant amount of attention in particular much is known about the description of fatou components besides the types of periodic fatou components that can occur for polynomials there also exist so called baker domains periodic components where all orbits converge to infinity as well as wandering domains in trying to find analogues of these one dimensional results it is not clear which higher dimensional transcendental maps to consider in this paper we find inspiration from the extensive work on the dynamics of complex henon maps we introduce the family of transcendental henon maps and study their dynamics emphasizing the description of fatou components we prove that the classification of the recurrent invariant fatou components is similar to that of polynomial henon maps and we give examples of baker domains and wandering domains

henon like mappings in c we study the dynamics of a class of nonalgebraic holomorphic diffeomorphisms topo logical analogues in the unit bidisk of complex h6non mappings in c2 in particular a dynamical degree is defined which is related to topological entropy and we construct stable unstable invariant currents and prove there is a unique mixing measure of maximal entropy with product structure 1 introduction one of the most famous 2 dimensional dynamical systems is smale s horseshoe it is a basic example of a simple map with complicated dynamics and has served as a guideline for the whole modern development of discrete time dynamical systems a very natural problem in this context is to understand the dynamical phenomena occuring when uniform hyperbolicity of the horseshoe degenerates this was a motivation for the study of the so called henon map

iteration and the solution of functional equations for functions analytic in the unit disk this paper considers the classical functional equations of schroeder f o p xf and abel f o p f 1 and related problems of fractional iteration where 9p is an analytic mapping of the open unit disk into itself the main theorem states that under very general conditions there is a linear fractional transformation 4 and a function a analytic in the disk such that a oa p and that with suitable normalization 4 and a are unique in particular the hypotheses are satisfied if 4p is a probability generating function that does not have a double zero at 0 this intertwining relates solutions of functional equations for t to solutions of the corresponding equations for d for example it follows that if 4p has no fixed points in the open disk then the solution space of f o p xf is infinite dimensional for every nonzero x although the discrete semigroup of iterates of t usually cannot be embedded in a continuous semigroup of analytic functions mapping the disk into itself we find that for each z in the disk all sufficiently large fractional iterates of p can be defined at z this enables us to find a function meromorphic in the disk that deserves to be called the infinitesimal generator of the semigroup of iterates of p if the iterates of p can be embedded in a continuous semigroup we show that the semigroup must come from the corresponding semigroup for 4 and thus be real analytic in t the proof of the main theorem is not based on the well known limit technique introduced by koenigs 1884 but rather on the construction of a riemann surface on which an extension of 9p is a bijection much work is devoted to relating characteristics of p to the particular linear fractional transformation constructed in the theorem

wandering domains in the iteration of entire functions fonctions entieres a domaines multiplement connexes de normalite ensembles de fatou julia pour des fonctions qui commuttent exemples de type d herman classes de fonctions entieres sans domaines d errance

the almost cyclicity of the fundamental groups of positively curved manifolds recall that a pure f structure is a kind of generalized torus action the main result asserts that if a compact positively curved manifold mn admits an invariant pure f structure such that each orbit has positive dimension then the fundamental group has a finite cyclic subgroup with index less than wn a constant depending only on n as an application we conclude that for all 0 1 the fundamental group of a pinched n manifold either has a cyclic subgroup with index less than wn or has order less than w n a constant depending only on n and in particular this substantially improves the main result in ro1

twisting and nonnegative curvature metrics on vector bundles over the round sphere a complete noncompact manifold m with nonnegative sectional curvature is diffeomorphic to the normal bundle of a compact submanifold s called the soul of m when s is a round sphere we show that the clutching map of this bundle is restricted this is used to deduce that there are at most finitely many isomorphism types of such bundles with sectional curvature lying in a fixed interval 0 we also examine the opposite question of how the twisting of the bundle limits the type of possible nonnegative curvature metrics on the bundle it turns out that if the bundle does not admit a nowhere zero section then the normal exponential map is necessarily a diffeomorphism onto m and the ideal boundary of m consists of a single point in their paper cg cheeger and gromoll raised the question of which vector bundles over the round sphere admit complete metrics with nonnegative sectional curvature the significance of this problem is that it attempts to determine to what extent a converse to the soul theorem holds recall that this theorem states that every open i e complete noncompact manifold m with nonnegative curvature km is diffeomorphic to a vector bundle over a compact totally geodesic submanifold s called a soul a natural question then is whether all such vector bundles admit complete metrics with km 0 in ow it was shown that when the soul is a bieberbach manifold nonnegative curvature metrics force the vanishing of the euler class of the vector bundle it follows that among oriented plane bundles over the torus only the trivial one admits such a metric the above case is fairly rigid a priori however since for example any bundle with curvature 0 over such a manifold must also admit a flat metric and the corresponding question for a simply connected base remains open an answer could provide insights on the topological structure of compact manifolds with nonnegative curvature since any open manifold with nonnegative curvature can after modifying its metric be isometrically 1991 mathematics subject classification primary 53c20

topology of riemannian submanifolds with prescribed boundary we prove that a smooth compact submanifold of codimension 2 immersed in r n 3 bounds at most finitely many topologically distinct compact nonnegatively curved hypersurfaces this settles a question of guan and spruck related to a problem of yau analogous results for complete fillings of arbitrary riemannian submanifolds are obtained as well on the other hand we show that these finiteness theorems may not hold if the codimension is too high or the prescribed boundary is not sufficiently regular our proofs employ among other methods a relative version of nash s isometric embedding theorem and the theory of alexandrov spaces with curvature bounded below including the compactness and stability theorems of gromov and perelman

the metric projection onto the soul we study geometric properties of the metric projection r m s of an open manifold m with nonnegative sectional curvature onto a soul s ir is shown to be c up to codimension 3 in arbitrary codimensions small metric balls around a soul turn out to be convex so that the unit normal bundle of s also admits a metric of nonnegative curvature next we examine how the horizontal curvatures at infinity determine the geometry of m and stu dy the structure of sharafutdinov lines we conclude with regularity properties of the cut and conjugate loci of m the resolution of the soul conjecture of cheeger and gromoll by perelinan showed that the structure of open manifolds with nonnegative sectional curvature is more rigid than expected orne of the key results in 14 is that the metric projection wf m s which maps a point p in m to the point 7r p in s that is closest to p is a riemannian submersioni perelman also observed that this map is at least of class c1 and later it was shown in 9 that 7w is at least c2 and c at almnost every point the existence of such a riemannian submersion combined with the restriction on the sign of the curvature suggests that the geometry of m inust be special the purpose of this paper is to illustrate this in several different directions in fact we show that most natural geometric objects classically used in the study of these spaces are intrinsically related to the structure of the map w the paper is essentially structured as follows after introducing notation and recalling some basic results in sectioni 1 we establish in section 2 the existence of maps similar to w for any submanifold of m homologous and isometric to the soul and use this to prove the existence of convex tubular neighborhoods for them in particular this also proves theorem 2 5 the unit normal bundle vti s admits a metric with nonnegative sectional curvature as a consequence open manifolds with nonnegative curvature provide examples of compact manifolds with the same lower curvatuire bound see also 81 theorem 2 5 also implies that nontrivial plane bundles over a bieberbach maanifold do not admit nonnegatively curved metrics thereby providing a shorter proof of the main result in 13 we are grateful to the referee for pointing out this fact to us properties such as these are established via stanidard submersion techniques but the fact that 7r is not known to be smooth everywhere prevents us fromi using these techniques to study the geometry of m far away from the soul one way around received by the editors august 18 1997 1991 mathematics subject classification primary 53c20

finiteness theorems for nonnegatively curved vector bundles we prove several finiteness theorems for the normal bundles to souls in nonnegatively curved manifolds more generally we obtain finiteness results for open riemannian manifolds whose topology is concentrated on compact domains of bounded geometry

obstructions to nonnegative curvature and rational homotopy theory we establish a link between rational homotopy theory and the problem which vector bundles admit a complete riemannian metric of nonnegative sectional curvature as an application we show for a large class of simply connected nonnegatively curved manifolds that if c lies in the class and t is a torus of positive dimension then most vector bundles over c x t admit no complete nonnegatively curved metrics

nonexistence of wandering domains for strongly dissipative infinitely renormalizable henon maps at the boundary of chaos this article generalizes the nonexistence of wandering domains from unimodal maps to infinitely period doubling renormalizable henon like maps in the strongly dissipative area contracting regime this solves an open problem proposed by van strien discrete contin dyn syst 27 2 557 588 2010 and lyubich and martens invent math 186 1 115 189 2011 we partition the phase space of a henon like map into two regions the good region and the bad region the good region is where the method of proof for unimodal maps applies to henon like maps while the bad region is where serious difficulties occur these difficulties are resolved by the two row lemma an inequality that relates the contraction of areas to the contraction of bad regions after analyzing the competition of the two types of contraction we show that the case of bad regions happens at most finitely many times and complete the proof as an application the theorem enriches our understanding of the topological structure of the heteroclinic web the union of the stable manifolds of periodic orbits forms a dense set in the domain

dubins problem on surfaces ii nonpositive curvature let m be a complete connected two dimensional riemannian manifold with nonpositive gaussian curvature k we say that m satisfies the unrestricted complete controllability ucc property for the dubins problem if the following holds given any p 1 v 1 and p 2 v 2 in tm there exists for every eps 0 a curve gamma in m with geodesic curvature smaller than eps such that gamma connects p 1 to p 2 and for i 1 2 dot gamma is equal to v i at p i property ucc is equivalent to the complete controllability of a family of control systems of dubins type parameterized by eps it is well known that the poincare half plane does not verify property ucc in this paper we provide a complete characterization of the two dimensional nonpositively curved manifolds m with either uniformly negative or bounded curvature that satisfy property ucc more precisely if sup mk infty we show that ucc holds if and only if i m is of the first kind or ii the curvature satisfies a suitable integral decay condition at infinity

riemannian submersions from compact four manifolds we show that under certain conditions a nontrivial riemannian submersion from positively curved four manifolds does not exist this gives a partial answer to a conjecture due to fred wilhelm we also prove a rigidity theorem for riemannian submersions with totally geodesic fibers from compact four dimensional einstein manifolds

when strictly locally convex hypersurfaces are embedded in this paper we will prove hadamard stoker type theorems in the following ambient spaces mathcal m n times mathbb r where mathcal m n is a 1 4 pinched manifold and certain killing submersions e g berger spheres and heisenberg spaces that is under the condition that the principal curvatures of an immersed hypersurface are greater than some non negative constant depending on the ambient space we prove that such a hypersurface is embedded and we also study its topology

cohomogeneity one disk bundles with normal homogeneous collars we consider cohomogeneity one homogeneous disk bundles and adress the question when these admit a nonnegatively curved1 invariant metric with normal collar i e such that near the boundary the metric is the product of an interval and a normal homogeneous space if such a bundle is not the quotient of a trivial bundle then we show that its rank has to be in 2 3 4 6 8 moreover we give a complete classification of such bundles of rank 6 and 8 and a partial classification for rank 3

diffeomorphism type of the berger space so 5 so 3 we compute the eells kuiper invariant of the berger manifold so 5 so 3 and determine that it is diffeomorphic to the total space of an s 3 bundle over s 4 this answers a question raised by k grove and w ziller

crc handbook of combinatorial designs balanced incomplete block designs and t designs2 v k l designs of small orderbibds with small block sizet designs t 3steiner systemssymmetric designsresolvable and near resolvable designslatin squares mols and orthogonal arrayslatin squaresmutually orthogonal latin squares mols incomplete molsorthogonal arrays of index more than oneorthogonal arrays of strength more than twopairwise balanced designspbds and gdds the basicspbds recursive constructionspbd closurepairwise balanced designs as linear spacespbds and gdds of higher indexpbds frames and resolvabilityother combinatorial designsassociation schemesbalanced part ternary designsbalanced tournament designsbhaskar rao designscomplete mappings and sequencings of finite groupsconfigurationscostas arrayscoveringscycle systemsdifference familiesdifference matricesdifference sets abeliandifference sets nonabeliandifference triangle setsdirected designsd optimal matricesembedding partial quasigroupsequidistant permutation arraysfactorial designsfrequency squaresgeneralized quadranglesgraph decompositions and designsgraphical designshadamard matrices and designshall triple systemshowell designsmaximal sets of molsmendelsohn designsthe oberwolfach problemordered designs and perpendicular arraysorthogonal designsorthogonal main effect planspackingspartial geometriespartially balanced incomplete block designsquasigroupsquasi symmetric designs r l designsroom squaresself orthogonal latin squares sols sols with a symmetric orthogonal mate solssom sequences with zero autocorrelationskolem sequencesspherical t designsstarterstrades and defining sets t m s netstuscan squarest wise balanced designsuniformly resolvable designsvector space designsweighing matrices and conference matriceswhist tournamentsyouden designs generalizedyouden squaresapplicationscodescomputer science selected applicationsapplications of designs to cryptographyderandomizationoptimality and efficiency comparing block designsgroup testingscheduling a tournamentwinning the lotteryrelated mathematics and computational methodsfinite groups and designsnumber theory and finite fieldsgraphs and multigraphsfactorizations of graphsstrongly regular graphstwo graphsclassical geometriesprojective planes nondesarguesiancomputational methods in design theoryindex

curvature and symmetry of milnor spheres since milnor s discovery of exotic spheres mi one of the most intriguing problems in riemannian geometry has been whether there are exotic spheres with positive curvature it is well known that there are exotic spheres that do not even admit metrics with positive scalar curvature hi on the other hand there are many examples of exotic spheres with positive ricci curvature cf ch1 he po and na and this work recently culminated in wr where it is shown that every exotic sphere that bounds a parallelizable manifold has a metric of positive ricci curvature this includes all exotic spheres in dimension 7 so far however no example of an exotic sphere with positive sectional curvature has been found in fact until now only one example of an exotic sphere with nonnegative sectional curvature was known the so called gromoll meyer sphere gm in dimension 7 as one of our main results we prove

classification of recurrent domains for some holomorphic maps abstract riemann surface as in theorem 1 2 we show first that f is not constant on any component fl v l 0 hence we can assume using a diagonal process that for a subsequence fm fm h and h ld on ut of p c q g2 h q q consequently u of n is closed in t2 since fro ld on zn and since f is not of finite order see fm then f is conjugate to an irrational rotation on the riemann surface e we then show as in theorem 2 3 of fs3 that z is contained in f2 so e is closed in i2 the rest of the argument is as in theorem 1 2 or as in fs3 theorem 2 3 references b bbd bs cg fm fs1 fs2 fs3 fs4 g hi ko kr m in rr u beardon a iteration of rational functions springer verlag 1991 barrett d bedford e dadok j t actions on holomorphically separable complex manifolds math z 202 1989 65 82 bedford e smillie j polynomial diffeomorphisms of 2 ii amer j math 4 1991 657 679 carleson l gamelin t complex dynamics springer verlag 1993 friedland s milnor j dynamical properties of plane automorphisms ergodic theory and dynamical systems 9 1989 67 99 fornaess j e sibony n complex dynamics in higher dimension i asterisque 222 1994 201 231 fornaess j e sibony n complex dynamics in higher dimension ii to appear in ann math studies fornaess j e sibony n complex henon mappings in 2 and fatou bieberbach domains duke math j 65 1992 345 380 fornaess j e sibony n holomorphic dynamical systems to appear gavosto e to appear herman m recent results on some open questions on siegel s linearization kobayashi s hyperbolic manifolds and holomorphic mappings new york marcel dekker 1970 kruzhilin n g holomorphic automorphisms of hyperbolic reinhardt do mains math ussr izvestya 32 1989 15 38 milnor j dynamics in one complex variable introductory lectures suny stony brook institute for mathematical sciences preprint 1990 5 narashiman r several complex variables univ chicago press 1971 rosay j p rudin w holomorphic maps from ie n to c n tams 310 1988 47 86 ueda t fatou set in complex dynamics in projective spaces preprint

the cut off covering spectrum we introduce the r cut off covering spectrum and the cut off covering spectrum of a metric space or a riemannian manifold the spectra measure the sizes of localized holes in the space and are defined using covering spaces called covers and r cut off covers they are investigated using 6 homotopies which are homotopies via grids whose squares are mapped into balls of radius on locally compact spaces we prove that these new spectra are subsets of the closure of the length spectrum we prove the r cut off covering spectrum is almost continuous with respect to the pointed gromov hausdorff convergence of spaces and that the cut off covering spectrum is also relatively well behaved this is not true of the covering spectrum defined in our earlier work which was shown to be well behaved on compact spaces we close by analyzing these spectra on riemannian manifolds with lower bounds on their sectional and ricci curvature and their limit spaces

strongly nonnegative curvature we prove that all currently known examples of manifolds with nonnegative sectional curvature satisfy a stronger condition their curvature operator can be modified with a 4 form to become positive semidefinite

conditions for nonnegative curvature on vector bundles and sphere bundles this paper addresses cheeger and gromoll s question of which vector bundles admit a complete metric of nonnegative curvature and relates their question to the issue of which sphere bundles admit a metric of positive curvature we show that any vector bundle which admits a metric of nonnegative curvature must admit a connection a tensor and a metric on the base space which together satisfy a certain differential inequality on the other hand a slight sharpening of this condition is sufficient for the associated sphere bundle to admit a metric of positive curvature our results sharpen and generalize walschap and strake s conditions under which a vector bundle admits a connection metric of nonnegative curvature

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