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This is the first of two papers in which we investigate the properties of the displacement functions of automorphisms of free groups (more generally, free products) on Culler-Vogtmann Outer space and its simplicial bordification - the free splitting complex - with respect to the Lipschitz metric. The theory for irreducible automorphisms being well-developed, we concentrate on the reducible case. Since we deal with the bordification, we develop all the needed tools in the more general setting of deformation spaces, and their associated free splitting complexes. In the present paper we study the local properties of the displacement function. In particular, we study its convexity properties and the behaviour at bordification points, by geometrically characterising its continuity-points. We prove that the global-simplex-displacement spectrum of $Aut(F_n)$ is a well-ordered subset of $\mathbb R$, this being helpful for algorithmic purposes. We introduce a weaker notion of train tracks, which we call {\em partial train tracks} (which coincides with the usual one for irreducible automorphisms) and we prove that, for any automorphism, points of minimal displacement - minpoints - coincide with the marked metric graphs that support partial train tracks. We show that any automorphism, reducible or not, has a partial train track (hence a minpoint) either in the outer space or its bordification. We show that, given an automorphism, any of its invariant free factors is seen in a partial train track map. In a subsequent paper we will prove that level sets of the displacement functions are connected, and we will apply that result to solve certain decision problems., 50 pages. Originally part of arXiv:1703.09945 . We decided to split that paper following the recommendations of a referee. Updated subsequent to acceptance by Transactions of the American Mathematical Society

Let ( M , ω ) (M, \omega ) be a connected, compact 6-dimensional symplectic manifold equipped with a semi-free Hamiltonian S 1 S^1 action such that the fixed point set consists of isolated points or surfaces. Assume dim H 2 ( M ) > 3 H^2(M)>3 . In an earlier paper, we defined a certain invariant of such spaces which consists of fixed point data and twist type, and we divided the possible values of these invariants into six “types”. In this paper, we construct such manifolds with these “types”. As a consequence, we have a precise list of the values of these invariants.

We prove that any infinite group interpretable in a separably closed field F of finite Ersov-invariant is definably isomorphic to an F-algebraic group. Using this result we show that any infinite field K interpretable in a separably closed field F is itself separably closed; in particular, in the finite invariant case K is definably isomorphic to a finite extension of F . This paper answers a question raised by D. Marker, whose help and guidance made this work possible. I would like to thank A. Pillay for helpful discussions, and E. Hrushovski for pointing out mistakes in the first version of this paper.

We show that under certain assumptions, the measurable cohomology class of the linear holonomy cocycle of a foliation yields information about the characteristic classes of the foliation. Combined with the results of a previous paper, this yields vanishing theorems for characteristic classes of certain actions of lattices in higher rank semisimple Lie groups. Let F be a discrete group acting by diffeomorphisms on a smooth compact manifold M. Associated to this action are certain characteristic classes in H (F, R), which are constructed as characteristic classes of a natural foliation associated to the group action. The action of F on M induces an action of F on the principal frame bundle P(M) of M and the characteristic classes of the action can be interpreted as obstructions to the existence of invariant geometric structures on M, i.e., principal subbundles of P(M) invariant by the F-action. If F is a lattice in a higher rank semisimple Lie group, then F has strong rigidity properties (see, e.g., [M, Z1]). In an earlier paper [S], we showed, using techniques from ergodic theory, that for a certain class of F-actions there is always an invariant measurable reductive geometric structure, i.e., a measurable principal subbundle with reductive structure group, which is invariant by the F-action. Moreover, the noncompact semisimple part of this reductive group is locally isomorphic to a semisimple factor of the ambient Lie group of F [Zi]. Zimmer [Z3] recently proved this result for a large class of actions (which does not a priori include the class of actions considered in [S]). A natural question is whether these results remain true in the smooth category. The purpose of this paper is to show that the characteristic classes, which obstruct a smooth geometric structure, vanish in the presence of a measurable geometric structure. Explicitly, we have Main Theorem. Let (M, Y) be a codimension n, C 2-foliated manifold and suppose that the linear holonomy cocycle is measurably equivalent to a locally tempered cocycle taking values in a subgroup H c GL(n, R) which is stable under transpose. Then the Weil homomorphism X: H'(g[(n), 0(n)) -Hc(M, Yi) Received by the editors February 24, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 57R32; Secondary 57S20. Research partially supported by an Alfred P. Sloan Dissertation Fellowship. ? 1991 American Mathematical Society 0002-9947/91 $1.00 + $.25 per page

In this paper the Kobayashi metric on a domain in Cn is used to define a new function space. Elements of this space belong to a nonisotropic Lipschitz class. It is proved that if f is holomorphic on the domain and in the classical Lipschitz space A. then in fact f is in the new function space. The result contains the original result of Stein on this subject and provides the optimal result adapted to any domain. In particular, it recovers the Hartogs extension phenomenon in the category of Lipschitz spaces. This paper is part of a program to work out the function theory-especially the harmonic analysis-of domains in Cn in terms of invariant metrics and related constructs. The point is that, whereas in one complex dimension the principal objects of study are functions, it turns out that in several complex dimensions the principal objects of study should be domains. The correct context in which to perceive the differences between domains seems to be that of metric geometry. The present paper is concerned with Lipschitz spaces, and draws its inspiration from [ST]. In [ST], Stein announced that holomorphic Lipschitz a functions on bounded C2 domains in Cn, n > 1, are in fact Lipschitz 2a in complex tangential directions; a detailed proof appears, for instance, in [KR3]. It turns out that this result is optimal only for strongly pseudoconvex domains (see [KR6] for improved results on some other domains), and our purpose here is to find a language in which to formulate the best result for any domain. The Eisenman-Kobayashi metric and volume construction provides the most natural language. For related work using the Eisenman-Kobayashi ideas, see [KR1, KR2, KM, GK]. A second point of the present work is to eliminate the arbitrary distinction between normal and tangential directions. All directions should be treated equally, and the metric should read off the variation in smoothness. An interesting by-product of the work is that it rediscovers the Hartogs extension phenomenon (or "Kugelsatz"-see [KR3]) in the category of Lipschitz functions. This feature is explored in ??2 and 4. ? 1 contains basic definitions and terminology, including the definition of a new Lipschitz class on domains in Cn. It also contains the statement of our Received by the editors October 12, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 32A10; Secondary 32A30. Work supported in part by the National Science Foundation. ? 1990 American Mathematical Society 0002-9947/90 $1.00 + $.25 per page

We investigate the Hermitian curvature flow (HCF) of left-invariant metrics on complex unimodular Lie groups. We show that in this setting the flow is governed by the Ricci-flow type equation $\partial_tg_{t}=-{\rm Ric}^{1,1} (g_t)$. The solution $g_t$ always exist for all positive times, and $(1 + t)^{-1}g_t$ converges as $t\to \infty$ in Cheeger-Gromov sense to a non-flat left-invariant soliton $(\bar G, \bar g)$. Moreover, up to homotheties on each of these groups there exists at most one left-invariant soliton solution, which is a static Hermitian metric if and only if the group is semisimple. In particular, compact quotients of complex semisimple Lie groups yield examples of compact non-K\"ahler manifolds with static Hermitian metrics. We also investigate the existence of static metrics on nilpotent Lie groups and we generalize a result in \cite{EFV} for the pluriclosed flow. In the last part of the paper we study HCF on Lie groups with abelian complex structures., Comment: 25 pages. Revised version. To appear in TAMS

In this paper we extend a recent breakthrough of Chen and Cheng on the existence of a constant scalar Kähler metric on a compact Kähler manifold to Calabi’s extremal metric. There are no new a priori estimates needed, but rather there are necessary modifications adapted to the extremal case. We prove that there exists an extremal metric with extremal vector V V if and only if the modified Mabuchi energy is proper, modulo the action of the subgroup in the identity component of the automorphism group which commutes with the flow of V V . We introduce two essentially equivalent notions, called reductive properness and reduced properness. We observe that one can test reductive properness/reduced properness only for invariant metrics. We prove that existence of an extremal metric is equivalent to reductive properness/reduced properness of the modified Mabuchi energy.

Unfortunately, there is a mistake in the proof of Theorem 3.3 of our paper entitled Hypersurfaces with constant mean curvature in the complex space which appeared in the Transactions of the AMS (Vol. 339 (1993), 685-702). In the proof of this theorem we applied Hopf's maximum principle, which holds for a class of hypersurfaces satisfying an elliptic PDE, to a one-parameter family of hypersurfaces with constant mean curvature (cmc) of Q5 (using the notations of the paper) obtained by reflecting an initial cmc hypersurface of Q5 on a one-parameter family of totally geodesic hypersurfaces of Q5. However, while we know that the initial hypersurface satisfies an elliptic PDE since it is invariant by the S' group of isometries of Q5 and has cmc, the reflections of the hypersurface do not satisfy the equation since, although they have cmc, they are not S1 invariant any more. Therefore, Hopf's maximum principle cannot be used and the proof, as it stands in the paper, is not correct. This mistake was pointed out to us by Professor J. Eschenburg. So far, we have not found a way to correct it and, in fact, this seems to be a difficult question.

In this paper, we show the convexity of the image of a moment map on a transverse symplectic manifold equipped with a torus action under a certain condition. We also study properties of moment maps in the case of transverse Kähler manifolds. As an application, we give a positive answer to the conjecture posed by Cupit-Foutou and Zaffran.

We consider the family of harmonic measures on a lamination $\mathcal{L}$ of a compact space $X$ by locally symmetric spaces $L$ of noncompact type, i.e. $L\simeq \Gamma_L\backslash G/K$. We establish a natural bijection between these measures and the measures on an associated lamination foliated by $G$-orbits, $\hat{\mathcal{L}}$ which are right invariant under a minimal parabolic (Borel) subgroup $B < G$. In the special case when $G$ is split, these measures correspond to the measures that are invariant under both the Weyl chamber flow and the stable horospherical flows on a certain bundle over the associated Weyl chamber lamination. We also show that the measures on $\hat{\mathcal{L}}$ right invariant under two distinct minimal parabolics, and therefore all of $G$, are in bijective correspondence with the holonomy-invariant ones., Comment: This paper has been accepted for publication in Transactions of the AMS

We study dynamical systems forced by a combination of random and deterministic noise and provide criteria, in terms of Lyapunov exponents, for the existence of random attractors with continuous structure in the fibres. For this purpose, we provide suitable random versions of the semiuniform ergodic theorem and also introduce and discuss some basic concepts of random topological dynamics., As one of the three main results of the first version was partially known (Y. Cao, 2006), we reorganized the material and changed the title of the submission in order to emphasize the other results and some new concepts introduced in this paper. (15 pages)

This paper deals with the classification of cyclic group actions on spheres using the Atiyah-Singer invariant and Reidemeister-type torsion. Our main tool is the computation of the group of relative homotopy triangulations of the product of a disk and a lens space. These results are applied to obtain lower bounds on the image of an equivariant J-homomorphism. Introduction. Smooth actions of finite groups on spheres which are semifree (the only isotropy groups are the identity subgroup and the whole group) and semilinear (the fixed set is a homotopy sphere) have received a considerable amount of attention in the past decade (see [20,21] and references there). Rothenberg [16] produced exact sequences relating groups of semilinear spheres to Wall groups and homotopy groups of classical groups and function spaces. Similar results were announced by Browder and Petrie [5] who carried out rational calculations using the Atiyah-Singer invariant (see the remarks after 5.4). Subsequently, Schultz refined and generalized these techniques to obtain much more detailed information [18,19,20]. Finally, Ewing's application of the G-signature theorem to semifree actions [7,8] was an important contribution to the analysis of Rothenberg's sequences (see 3.4, 5.3 below). In this paper, we study PL actions which are semifree and semilinear, as well as smooth near the fixed set. The advantage of this hybrid category is that in it explicit calculations with the Atiyah-Singer invariant can be carried out which allow classification of semilinear spheres. These are modeled on Wall's study of free piecewise linear actions of odd order cyclic groups on spheres [25, ?14E]. Out of these calculations come the invariants which detect elements in the image of an equivariant J-homomorphism. 1. Definitions and summary of results. Let G be a cyclic group of odd order q and let a = n D u be a representation of G which is the sum of the trivial n dimensional real representation and an orthogonal representation u such that G acts freely on S(u), the unit sphere of u. An a-manifold is a PL manifold with a semifree PL G-action together with the following additional structure: (1) A neighborhood of the fixed set is provided with a sliced concordance class of G-smoothings [11]. (2) The Received by the editors June 3, 1981. 1980 Mathematics Subject Classification. Primary 57S25; Secondary 57R67, 55Q50. 'This work was partially supported by NSF Grant MCS 77-18723 A04. ?1983 American Mathematical Society 0002-9947/82/0000-0717/$06.00

A linear action of an abelian group on a sphere generally contains a large family of invariant linear subspheres. In this paper the problem of finding invariant subspheres for more general smooth actions on homotopy spheres is considered. Classification schemes for actions with invariant subspheres are obtained; these are formally parallel to the classifications discussed in the preceding paper of this series. The realizability of a given smooth action as an invariant codimension two sub- sphere is shown to depend only on the ambient differential structure and an isotopy invariant. Applications of these results to specific cases are given; for example, it is shown that every exotic 10-sphere admits a smooth circle action. In our previous papers in this series (45,44), we have considered the theory of semifree actions on homotopy spheres as formulated by W. Browder and T. Petrie (10) and M. Rothenberg and J. Sondow (34). Specifically, in the first paper a method was presented for describing (at least formally) those exotic spheres admitting such semifree actions- a problem first posed explicitly by Browder in (3, Problem 1, p. 7) -and the seconid paper extended the whole theory to handle certain actions that are not semifree. This paper will treat another problem posed in Browder's paper (3, Problem 3) regarding invariant subspheres of homotopy spheres with group actions. One motivation for considering this question is that linear actions on spheres generally admit a great assortment of invariant linear subspheres (e.g., if the group is abelian and the dimension is much larger than the group's order), and from this viewpoint the existence of invariant subspheres reflects the extent to which an arbitrary smooth action resembles some natural linear model. In particular, this idea is central to the work of Browder and Livesay on free involutions (8) (compare also (3)). The existence of such subspheres is directly related to the realizability of actions as equivariant smooth suspensions, providing basic necessary conditions for such a realization. We shall also consider a dual problem in this paper; namely, the description of those group actions that can be smoothly equivariantly suspended. Questions of this sort first arose in the study of free involutions (16), and their close

Generalizing a construction of Banach from 1923 we obtain new criteria for the amenability of a locally compact group G. The relationship of these new criteria to known characterizations is then investigated, and in particular a formally strenghthened version of the von Neumann/Dixmier condition for amenability is established. 9. Introduction and notation. In 1923 Banach [2] showed how to construct translation invariant means on Ll(T) (where T is the circle) which do not agree with the Lebesgue integral in general but do agree on the subspace of Riemann integrable functions, and thereby resolved the "Probleme de la mesure" which had been open since its proposal in Lebesgue's "Leqons sur I'Integration" eighteen years earlier. Contained in Banach's construction naturally generalized to an arbitrary locally compact group G there is only one point of possible obstruction, and consequently when isolated and formalized one obtains a condition on G which is readily seen to be equivalent to amenability (in current terminology). It is rather curious that contained implicitly in what is perhaps the first paper on translation invariant means/amenability is a characterization of amenability which has apparently gone unnoticed for over 50 years. The purpose of the present paper is to describe and examine this condition, showing its relationship to known characterizations and to derive further consequences. In what follows G is a fixed arbitrary locally compact group with a fixed left Haar measure associated, Lp = LP(G) are the associated real Lebesgue spaces (1 < p < oo), and S = S(G) is the real vector space of all real measurable simple functions on G. Moreover, for any function f on G, LJ denotes left translation by a ((L

We describe a generalization of Thurston's original construction of pseudo-Anosov maps on a surface F of negative Euler characteristic. In fact, we construct whole semigroups of pseudo-Anosov maps by taking appropriate compositions of Dehn twists along certain families of curves; our arguments furthermore apply to give examples of pseudo-Anosov maps on nonorientable surfaces. For each self-map f: F -* F arising from our recipe, we construct an invariant "bigon track" (a slight generalization of train track) whose incidence matrix is Perron-Frobenius. Standard arguments produce a projective measured foliation invariant by f. To finally prove that f is pseudo-Anosov, we directly produce a transverse invariant projective measured foliation using tangential measures on bigon tracks. As a consequence of our argument, we derive a simple criterion for a surface automorphism to be pseudo-Anosov. Introduction. A homeomorphism p of a surface F is said to be pseudo-Anoosov if no iterate of p fixes any essential nonboundaryor puncture-parallel free homotopy class of simple curves in F. Examples of these homeomorphisms date back to the work of Nielsen (see [N and Gi]), but a systematic study of these maps was not undertaken until the work of Thurston [T1]. Anosov [A] studied maps of the torus which preserve two foliations of the torus by lines of irrational slope, and pseudo-Anosov maps on F similarly preserve a pair of foliations (with singularities). Pseudo-Anosov maps are by no means special; indeed, the monodromy of any nontorus fibred knot which is not a satellite is pseudo-Anosov [T4]. (Note that being pseudo-Anosov is a conjugacy invariant.) Moreover, these maps play an important role in the geometrization of three-manifolds; indeed, a mapping torus has hyperbolic structure if and only if the monodromy is pseudo-Anosov [T4]. In the original preprint [T1], there is described a construction of pseudo-Anosov maps which we will recall later. In this paper, we generalize Thurston's construction and give a recipe for constructing whole semigroups of pseudo-Anosov maps, many of which do not arise from Thurston's construction. Our recipe is also applicable to nonorientable surfaces, and we give examples of pseudo-Anosov maps in this setting. (In [T3], Thurston proved the existence of such, and [AY] gave the first explicit examples.) This paper is organized as follows. In ?1, we review the basic terminology and results on train tracks in surfaces and indicate the connection between measured train tracks and measured foliations. ?2 is devoted to tangential measure on bigon Received by the editors March 7, 1986 and, in revised form, June 17, 1987. 1980 Mathematics Subject (Cassification (1985 Revision). Primary 57N06, 57N50.

Introduction. We shall consider a certain class of functions invariant with respect to the substitutions of the congruence subgroup Fo(p) of the modular group r. By specializing these functions, we shall obtain classical identities in the analytical theory of numbers: E.g., the Ramanujan identities for partitions modulo 5, 7 and Mordell's identity for r(n). We shall also derive some new identities. These functions bear some resemblance to those considered by Rademacher in his paper [1 ](1) to prove the Ramanujan identities, certain modular equations, etc. The type of function considered, however, seems first to have been studied by Watson in his paper [2]. 1. Definitions, notations. (1.1) r is the full modular group; i.e., the group of 2 X2 matrices of determinant 1 with rational integral elements. (1.2) Fo(m) is the subgroup of r characterized as follows: The element

In this paper, we investigate the possibility of proving analogous theorems for meromorphic functions possessing deficient values (in the sense of R. Nevanlinna). The main interest of the results obtained lies in the fact that they provide partial answers to the three following questions. I. Under which conditions are deficiencies invariant under a change of origin? II. When are deficient values also asymptotic values? III. How does the presence of deficient values influence the gap structure of the Taylor expansion of an entire function? We leave aside questions II and III which will be treated in another paper [1]. We explain our notations in ?1 before stating our results in ?2. 1. Terminology and notations. The complex variable will be denoted by

In the article t A ne?w method of determining the differential parameters and invariants of qutadraitic differential quantics I have shown that the application of a certain symbolic method leads very readily to the formation of expressions remaining invariant with respect to the transformation of quadratic differential quantics. The presentation in that article was only a preliminary one and the work practically confined to the case of two independent variables. In my paper 4 Invariants and covariants of quadratic differential quantics qf n variables a more complete treatment was intended and the investigation applied throughout to the case of n variables, leaving aside, however, simultaneous invariant forms of more than one quantic. The present paper contains in ?? 1-6 and ? 8 essentially the content of the last mentioned paper; the greater parts of ? 5 and ? 8, and all the remaining articles are new, in particular the extensive use of covariantive differentiation.

The problem of the determination t of all linear transformations possessing an invariant quadratic form, is well kniown to be classic. It enjoyed the attention of EULER, CAYLEY and HERMITE, and reached a certain stag,e of completeness in the memoirs of FROBENIUS, Voss,? LINDEMANN| and LOEWY.? The investigations of CAYLEY and HERMITE were confined to the g-eneral tranlsformation, FROBENIUS then determined all proper transformations, and finally the problem was completely solved by LINDEMANN and LOEWY, and simplified by Voss. The present paper attacks the problem from an altogether different point, the fundamental idea being that of building up any such transformation from simple elements. The primary transformation is taken to be central rexfection in the quadratic locus defined by settinig the given form equal to zero. This transformation is otherwise called in three dimensions, point-plane reflection,point and plane being pole and polar plane with respect to the fundamental quadric. In this way, every linear transfornmatioln of the desired form is found to be a product of central reflections. The maximum number necessary for the most general case is the number of variables. Voss, in the first memoir cited, proved this theorem for the general transformation, assuming the latter given by the equations of CAYLEY. In the present paper, however, the theorem is derived synithetically, and from this the analytic formi of the equations of transformation is deduced.

The theory of invariants originally confined itself to forms involving a single set of homogeneous variables; but recent investigations, geometric as well as algebraic, have proved the importance of the study of forms in any number of sets of variables. In passing from the theory of the simple to the theory of the multiple forms, an entirely new feature presents itself: in the latter case the linear transformations which are fundamental in the definition of invariants may be the same for all the variables or they may be distinct, i. e., the sets of variables involved may be cogredient or digredient. Multiple forms thus have two distinct invariant theories, a cogredient and a digredient. The object of this paper is to study the relations between these two theories in the case of forms involving any number of binary variables. Geometrically, such a form may be regarded as establishing a correspondence between the elements of two or more linear manifolds; in the digredient theory the latter are considered as distinct, thus undergoing independent projective transformations, while in the cogredient theory the linear manifolds are considered to be superposed, thus undergoing the same projective transformation. The first part of the paper, ?? 1-5, is devoted to the double forms. The extension of the results is made first, for convenience of presentation, to the triple forms in ? 6, and then to the general case in ? 7. The case of the double binary forms is perhaps the most interesting geometrically. In addition to the general interpretation by means of an algebraic correspondence between two manifolds, such a form may be interpreted as an algebraic curve on a quadric surface, or as a plane algebraic curve from the view point of inversion geometry. In the former of these special interpretations the two binary variables are the (homogeneous) parameters of the two sets of generators on the quadric, while in the latter they are the parameters of the two sets of minimal lines in the plane. These interpretations suggest the

This paper is devoted to the study of certain families of orbits in the problem of three bodies, which lie in the neighborhood of known orbits of simple types, but in which the masses are arbitrary and fixed. The classical method of variation of a parameter which enters into the equations of motion cannot, therefore, be employed; instead, extensive use is made of methods in which the notion of invariant relationt plays a prominent role. In the first two paragraphs are developed certain implications regarding the equations of variation, which result from the existence of a set of invariant relations satisfied by the generating solution of a differential system, following which certain properties of isosceles triangle solutions with axis of symmetry in the problem of three bodies are established. The latter part of the paper is given up to the study of plane orbits which lie in the neighborhood of the straight line solutions, or in the neighborhood of the equilateral triangle solutions, respectively; in this study a simple reduction of the equations of motion in the plane is obtained, which is found especially convenient. Many of the results obtained concerning these last solutions have been given by D. Buchanan,: but the method employed here has made it possible to go farther in certain respects. Certain related questions of stability will be taken up in a later paper.

(1) T1' =ATJA, T2' = AT2A where Af denotes the transpose of A. Thus the ranks of T1 and T2 are invariant, and if ; is a real field, so are the signatures. Other elementary properties of these matrices were discussed and their occurrence in the literature noted. In the first part of the present paper the behavior of T1 under transformation of basis is used to establish the existence for every algebra with a principal unit of a normal basis of simple form. This normal basis has a cyclic property generalizing that of the familiar basis 1, i, j, k for quaternions. By means of this normal basis several new theorems in the theory of semi-simple algebras are obtained, e.g., the fact that T1 and T2 are identical, and that the first and seconid characteristic functions are identical. In the second part of the paper (?4 et seq.) the discriminant matrices of a direct sum, direct product and complete matric algebra are investigated. 2. The normal basis. Let us now assume that e1 is a principal unit so that

0. Introduction and summary. The original and principal problem studied in this paper is that of finding conditions to be satisfied by a positive "weight function" w on a locally compact group in order that the set of functions f such that f Pw is integrable (relative to left Haar measure) shall be stable under left (or right) translations. Related to this is the question which asks when the set of f such that fw is integrable shall form an algebra under convolution. A good deal of the preliminary matter related to the first problem can and will be formulated in a more general setting. Accordingly throughout this paper, and subject to specialization adopted in various sections, X denotes a locally compact space and yX0 a fixed positive Radon measure on X. S is a set of homeomorphisms s: x->s x of X onto itself. For our purposes there is no generality lost by assuming that S is a semigroup under composition and that it contains the identity map 1: X--X. We assume that , is relatively invariant under S, i.e. that there exists a real-valued function A on S such that

1. In view of the recent work on topological groups it is natural to consider the situation which arises when such groups act as transformation groups on various types of spaces. Such a study is begun here from the point of view of almost periodic transformation groups, the definition of which is suggested by von Neumann's paper on almost periodic functions in a group.t Compact topological transformation groups are a special case of almost periodic transformation groups, at least for a rather wide class of spaces. The paper concerns itself chiefly with the nature of the minimal closed invariant sets of such groups. There are some results for general spaces but the main results are for Euclidean spaces and more particularly for threedimensional Euclidean spaces. One of the most interesting theorems states that if a compact one-dimensional group acts on three-space in such a way that its orbits are uniformly bounded in diameter, then every point of the space is fixed under the group, so that if such a group is to act in a nontrivial manner the diameters of its orbits must be unbounded. Under some restrictions a similar theorem is proved for one-parameter almost periodic groups. Furthermore it is shown that for this latter class of groups, many of the orbits must actually be simple closed curves if they have one-dimensional closures. 2. The group considered here will be denoted by G. It will be subjected to various conditions as the occasion demands but it will always be Abelian. In case it is the group of real numbers, it will be spoken of as a one-parameter group; in case it is the real numbers reduced modulo one, it will be spoken of as the circle group. The space on which the group acts will be denoted by R. It will be specialized in various ways, but in any case it will always be a locally compact metric space. If x and y are two points of R, the distance between them will be denoted by d(x, y).

In a recent paper Dr. ROSENBLATT gives two interesting examples of algebraic surfaces which are invariant under an infinite discontinuous group of birational transformations and at the same time are not envelopes of quadric surfaces.t In an earlier paper I mentioned 1: that all surfaces belonging to such groups which have thus far been noticed were examples of surfaces defining an ordinaxy elliptic ( 2, 2 ) correspondence. Practically all the memoirs bearing on the problem are mentioned in the articles just named. The surfaces discussed by DR. ROSENBLATT have a pencil of elliptic curves, and the transformations are expressed by a linear transformation of the parameters u, v of elliptic functionsn in terms of which the coordinates of a point on either surface can be rationally expressed. The treatment is transcendental, the transformation is defined only for points on the surface and no explanation of the geometric meaning of the transformation is given. It is a curious fact that these transformations are birational for all space, that they have a simple geometric interpretation in terms of the (2 X 2 ) correspondence, and that surfaces of any order or of as high a geometric genus as may be desired can be constructed which are invariant under this group, or a larger group under which that discussed by Dr. Rosenblatt is a subgroup of infinite index. Consider the surface whose equation is of the form

A number of recent books deal with the theory of groups of linear transformations and its connection with the theory of algebras(1). Most of the work has been restricted to the case of completely reducible systems or, in other words, to semisimple algebras. There are, however, a number of questions which make it desirable not to neglect the other case. The aim of this and a following paper is a study of such not completely reducible systems, in particular of their regular representations. It appeared necessary to start again right from the beginning of the theory, in order to add a number of remarks to well known results and methods(2). The coefficients of the matrices in this paper are taken from an arbitrary division ring K (=skew field or noncommutative field K). This is a generalization of the ordinary theory which does not always work smoothly. For instance, the (left) rank of a ring of matrices t is not invariant under similarity transformation. This implies that similar rings 2L and ?1, may have different regular representations. Yet it is possible to derive a number of results which, in the case of a commutative K, imply the fundamental theorems of Frobenius, Burnside, Loewy, I. Schur and Wedderburn. Sections 1 and 2 deal with a number of group-theoretical remarks. The first of these are concerned with the Jordan-Holder theorem. The connection between two composition series is studied more closely, and it is proved that sets of residue systems can be chosen such that they can be used in either composition series. Further, the upper and lower Loewy series of a group are studied. It is shown that the ith factor groups in both have a common constituent. This implies the theorem of Krull and Ore(3) that both series have the same length. In Section 3, the necessary tools from the theory of matrices are described briefly. The following two sections contain an application of the group-theoretical methods to the study of the irreducible and the Loewy constituents of a set of matrices. In Section 6, a number of further remarks are added, for instance a generalization of a theorem of A. H. Clifford(').

In the metric differential geometry of surfaces in ordinary space, two surfaces are said by Bianchi to be associatet if the tangent planes at corresponding points are parallel and if the asymptotic curves on either surface correspond to a conjugate net on the other. It is the purpose of this paper to develop a projective generalization of the relation of associateness of surfaces. Since associate surfaces are parallel in the metric sense, it will first be necessary to provide a projectively defined substitute for the property of metric parallelism. We shall employ as the basis of our study in this paper a projective generalization of euclidean parallelism of surfaces which the author has developed in his Chicago doctoral dissertation. In ?2, after stating a definition of projective parallelism of surfaces and briefly explaining this idea, we introduce a canonical form of our system of differential equations employed in the study of projectively parallel surfaces in ordinary space. In ?3 we formulate a definition of projectively associate surfaces and investigate to some extent their properties and relations. A more general type of associateness which may be conveniently termed modified projective associateness is introduced in ?4, and a somewhat different canonical form of our system of differential equations is employed in its study. Finally, in ?5, we consider a rather general completely integrable system of partial differential equations, namely, the system for two surfaces in the general analytic one-to-one point correspondence in ordinary projective space S3, and a group of transformations that leaves this configuration invariant. We then reduce this system of equations to a new canonical form, and employ it to continue briefly the study of modified projective associateness introduced in the preceding section.

Introduction. In this paper, we shall begin the study of the invariant theory of the most general irregular analytic arc in the geometry based on the infinite group G of arbitrary regular point transformations. Our results are valid for the group of real point transformations of the real plane; or for the group of complex point transformations of the complex plane. Kasner has developed the corresponding theory in the conformal geometry of the complex plane(1). The present paper opens up a new aspect of restricted topology. Our subject is then the equivalence theory of a single arc or curve. When can one analytic arc be converted into another analytic arc by an arbitrary regular point transformation of the plane? It is apparently implied, in the current literature, that there is no problem here. For any curve (it is implied) can be converted into any other, in particular, into the x-axis. But this is based on the assumption that the arcs are regular. If we give up this assumption, we have actual problems which certainly seem worthy of treatment. Our subject is therefore the invariant theory of a general irregular analytic arc under the group G of arbitrary regular point transformations. More exactly, the configuration we shall discuss is not simply an analytic arc but rather that arc together with a specific point of the arc. This compound configuration we shall term an analytic element. It consists of a pointthe base point which shall be taken as the origin throughout this paper-and an analytic arc through the point. It may be described also as a differential element of infinite order(2). The most general analytic element, if the given point o is taken as origin, is represented by writing x and y as integral power series in a parameter t without the constant terms. If the parameter t is eliminated, then y is found as a series in x which may proceed according to integral or fractional powers

The problem about to be discussed may be looked upon as a generalization of the classical problem solved by Sturm in 1829 with regard to the real roots of a polynomial.t In Sturm's theorem it is shown how the number of distinct roots of a single polynomial which fall within a given real interval mnay be determined through a process rational in the coefficients. In the present paper we shall study a system of two or more polynomials in a single variable, and our aim will be to develop a rational process by which the order of succession of the roots of the several polynomials in a given real interval may be discovered. Let us confine ourselves, at least for the present, to the case where the endvalues of the interval are not roots and where the polynomials have only simple roots in the interval. It is clear that in such a case Sturm's theorem determines the only relations of the real roots of a single polynomial to the interval that remain invariant under continuous transformation of the real number system into itself. A similar remark applies to the theory about to be developed with respect to the real root system of several polynomials; so that we are undertaking the study of a problem which, from the point of view of a onedimensional analysis situs, may be said to be the fundamental problem of the system under consideration. The greater part of the paper will be devoted to the case of two polynomials. If the roots of the first are denoted generally by a and those of the second by f, and if we write down the roots within the interval considered in increasing numerical order (as aaa af3f3c43oa), the O's effect a certain partition of the a's (in the present case into groups of 3, 0, 1, O, O, 1, O,). The solution of the problem will consist in the determination of the numbers of a's in the successive groups.

We study differential operators D which commute with a fixed normal elliptic operator E on a compact manifold M. We use eigenfunction expansions relative to E to obtain simple conditions giving global hypoellipticity. These conditions are equivalent to D having parametrices in certain spaces of functions or distributions. An example is given by M = compact Lie group and and E = Casimir operator, with D any invariant differential operator. The connections with global subelliptic estimates are investigated. 0. Introduction. We say a differential operator D on a manifold M is globally hypoelliptic (GH) if when D/= g (with f E '(M), g E C"c(M)) then / E coc (M). We begin this paper by recalling some Fourier analysis relative to an elliptic operator E on a compact manifold M, and apply this to obtain simple conditions on the rate of growth of the Fourier transform (relative to E) of D which are equivalent to (GH). The growth conditions are interpreted as global solvability conditions. We apply these theorems to the case: M = compact homogeneous space, E = invariant Laplace-Beltrami operator, and D = any invariant differential operator. Some new examples are discussed. We note that global hypoellipticity seems to be quite directly connected with questions of number theory-unlike the (analytically) more delicate questions of local hypoellipticity. We connect the eigenfunction estimates of E-Fourier analysis with global subelliptic estimates. Almost every result in this paper can be extended to differential operators on vector bundles (see Wallach [11 I for the basic ideas). We thank Carl Hoel and William Sweeney for patiently teaching us about differential equations. We also thank Richard Bumby for suggesting the use of Pell's equation in ?3. 1. Fourier analysis relative to an elliptic operator. Let M be a compact manifold without boundary of dimension n with a fixed volume element dv. Let E be an elliptic, normal (EE =E E) differential operator of order e on M. We Received by the editors July 10, 1972. AMS (MOS) subject classifications (1970). Primary 35H05; Secondary 43A80. (1) Partially supported by NSF GP-20647. (2) Partially supported by Alfred P. Sloan Fellowship. Copyright

This paper initiates a series of papers which will reexamine some problems and results of classical invariant theory, within the framework of modern first-order logic. In this paper the notion that an equation is of invariant significance for the general linear group is extended in two directions. It is extended to define invariance of an arbitrary first-order formula for a category of linear transformations between vector spaces of dimension n. These invariant formulas are characterized by equivalence to formulas of a particular syntactic form: homogeneous formulas in determinants or “brackets". The fuller category of all semilinear transformations is also introduced in order to cover all changes of coordinates in a projective space. Invariance for this category is investigated. The results are extended to cover invariant formulas with both covariant and contravariant vectors. Finally, Klein’s Erlanger Program is reexamined in the light of the extended notion of invariance as well as some possible geometric categories.

Introduction. Among the topological invariants of a space X certain spaces have frequently been found valuable. The space of all continuous functions on X and the space of mappings of X into a circle are noteworthy examples. It is the purpose of this paper to study two particular invariant spaces associated with a compact metric continuum X; namely, 2X, which consists of all closed nonvacuous subsets of X, and C(X), which consists of closed connected nonvacuous subsets('). The aim of this study is twofold. First, we wish to investigate at length the topological properties of the hyperspaces, and, second, to make use of their structure to prove several general theorems. If X is a compact metric continuum it is known that: 2X is Peanian if X is Peanian [7 ], and conversely [8]; 2x is always arcwise connected [1 ]; 2X is the continuous image of the Cantor star [4]; if X is Peanian, each of 2x and C(X) is contractible in itself [9]; and if X is Peanian, 2x and C(X) are absolute retracts [10]. In ??1-5 of this paper further topological properties are obtained. In particular: 2x has vanishing homology groups of dimension greater than 0, both hyperspaces have very strong higher local connectivity and connectivity properties-including local p-connectedness in the sense of Lefschetz for p > 0, and, the question of dimension is resolved except for the dimension of C(X) when X is non-Peanian. All of the results of the preceding paragraph for 2x are shown simultaneously for 2x and (X) in the course of the development. In ?6 a characterization of local separating points in terms of C(X) is obtained and a theorem of G. T. Whyburn deduced. In ?7 it is shown that for a continuous transformation f(X) = Y we may under certain conditions find XoCX, with XO closed and of dimension 0, such that f(Xo) = Y. In ?8 this result is utilized in the study of Knaster continua. In order that X be a Knaster continuum it is necessary and sufficient that C(X) contain a unique arc between every pair of elements. If there exist Knaster continua of dimension greater than 1 then there exist infinite-dimensional Knaster continua.

Let H H be a hypergroup with left Haar measure. The amenability of H H can be characterized by the existence of nets of positive, norm one functions in L 1 ( H ) L^1(H) which tend to left invariance in any of several ways. In this paper we present a characterization of the amenability of H H using configuration equations. Extending work of Rosenblatt and Willis, we construct, for a certain class of hypergroups, nets in L 1 ( H ) L^1(H) which tend to left invariance weakly, but not in norm. We define the semidirect product of H H with a locally compact group. We show that the semidirect product of an amenable hypergroup and an amenable locally compact group is an amenable hypergroup and show how to construct Reiter nets for this semidirect product. These results are generalized to Lau algebras, providing a new characterization of left amenability of a Lau algebra and a notion of a semidirect product of a Lau algebra with a locally compact group. The semidirect product of a left amenable Lau algebra with an amenable locally compact group is shown to be a left amenable Lau algebra.

K. Grove, L. Verdiani, B. Wilking and W. Ziller gave the first examples of cohomogeneity one manifolds which do not carry invariant metrics with non-negative sectional curvatures. In this paper we generalize their results to a larger family. We also classified all class one representations for a pair (G;H) with G/H some sphere, which are used to construct the examples.

This paper introduces new invariants for multiparameter dynamical systems. This is done by counting the number of points whose orbits intersect at time n under simultaneous iteration of finitely many endomorphisms. We call these points synchronization points. The resulting sequences of counts together with generating functions and growth rates are subsequently investigated for homeomorphisms of compact metric spaces, toral automorphisms and compact abelian group epimorphisms. Synchronization points are also used to generate invariant measures and the distribution properties of these are analysed for the algebraic systems considered. Furthermore, these systems reveal strong connections between the new invariants and problems of active interest in number theory, relating to heights and greatest common divisors

In this paper we use the relationship between conformal metrics on the sphere and horospherically convex hypersurfaces in the hyperbolic space for giving sufficient conditions on a conformal metric to be radial under some constraints on the eigenvalues of its Schouten tensor. Also, we study conformal metrics on the sphere which are invariant by a k-parameter subgroup of conformal diffeomorphisms of the sphere, giving a bound on its maximum dimension. Moreover, we classify conformal metrics on the sphere whose eigenvalues of the Shouten tensor are all constant (we call them isoparametric conformal metrics), and we use a classification result for radial conformal metrics which are solutions of some σk-Yamabe type problem for obtaining existence of rotational spheres and Delaunay-type hypersurfaces for some classes ofWeingarten hypersurfaces in Hn+1. © 2011 American Mathematical Society.

Patterns of invariant sets of interval maps are the equivalence classes of invariant sets under order-preserving conjugacy. In this paper we study forcing relations on patterns of invariant sets and reductions of interval maps. We show that for any interval map f f and any nonempty invariant set S S of f f there exists a reduction g g of f f such that g | S = f | S g|_S=f|_S and g g is a monotonic extension of f | S f|_S . By means of reductions of interval maps, we obtain some general results about forcing relations between the patterns of invariant sets of interval maps, which extend known results about forcing relations between patterns of periodic orbits. We also give sufficient conditions for a general pattern to force a given minimal pattern in the sense of Bobok. Moreover, as applications, we give a new and simple proof of the converse of the Sharkovskiĭ Theorem and study fissions of periodic orbits, entropies of patterns, etc.

In this paper, the Bando-Futaki invariants on hypersurfaces are derived in terms of the degree of the defining polynomials, the dimension of the underlying projective space, and the given holomorphic vector field. In addition, the holomorphic invariant introduced by Tian and Chen (Ricci Flow on K\"ahler-Einstein surfaces) is proven to be the Futaki invariant on compact K\"ahler manifolds with positive first Chern class., Comment: 44 pages

In their famous paper, Schweizer and Smítal introduced the definition of a distributionally chaotic pair and proved that the existence of such a pair implies positive topological entropy for continuous mappings of a compact interval. Further, their approach was extended to the general compact metric space case. In this article we provide an example which shows that the definition of distributional chaos (and as a result Li-Yorke chaos) may be fulfilled by a dynamical system with (intuitively) regular dynamics embedded in R 3 \mathbb {R}^3 . Next, we state strengthened versions of distributional chaos which, as we show, are present in systems commonly considered to have complex dynamics. We also prove that any interval map with positive topological entropy contains two invariant subsets X , Y ⊂ I X,Y \subset I such that f | X f|_X has positive topological entropy and f | Y f|_Y displays distributional chaos of type 1 1 , but not conversely.

We introduce new biholomorphic invariants for real-analytic hypersurfaces in C 2 \mathbb {C}^2 and show how they can be used to show that a hypersurface possesses few automorphisms. We give conditions, in terms of the new invariants, guaranteeing that the stability group is finite, and give (sharp) bounds on the cardinality of the stability group in this case. We also give a sufficient condition for the stability group to be trivial. The main technical tool developed in this paper is a complete (formal) normal form for a certain class of hypersurfaces. As a byproduct, a complete classification, up to biholomorphic equivalence, of the finite type hypersurfaces in this class is obtained.

In [GM1], we defined a G_R-K_C invariant subset C(S) of G_C for each K_C-orbit S on every flag manifold G_C/P and conjectured that the connected component C(S)_0 of the identity would be equal to the Akhiezer-Gindikin domain D if S is of nonholomorphic type. This conjecture was proved for closed S in [WZ2,WZ3,FH,M4] and for open S in [M4]. It was proved for the other orbits in [M5] when G_R is of non-Hermitian type. In this paper, we prove the conjecture for an arbitrary non-closed K_C-orbit when G_R is of Hermitian type. Thus the conjecture is completely solved affirmatively., 15 pages

The DoCarmo-Wallach moduli space parametrizing spherical minimal immersions of a Riemannian manifold M is a compact convex body in a linear space of tracefree symmetric endomorphisms of an eigenspace of M. In this paper we define and study a sequence of metric invariants σ m , m > 1, associated to a compact convex body L with base point O in the interior of L. The invariant σ m measures how lopsided L. is in dimension m with respect to 0. The results are then appplied to the DoCarmo-Wallach moduli space. We also give an efficient algorithm to calculate σ m for convex polytopes.

Let E be a ring spectrum for which the E-Adams spectral sequence converges. We define a variant of Mahowald's root invariant called the `filtered root invariant' which takes values in the E_1 term of the E-Adams spectral sequence. The main theorems of this paper concern when these filtered root invariants detect the actual root invariant, and explain a relationship between filtered root invariants and differentials and compositions in the E-Adams spectral sequence. These theorems are compared to some known computations of root invariants at the prime 2. We use the filtered root invariants to compute some low dimensional root invariants of v_1-periodic elements at the prime 3. We also compute the root invariants of some infinite v_1-periodic families of elements at the prime 3., Comment: 63 pages, 4 figures, to appear in Trans. AMS

The transverse (saturated) Lusternik-Schnirelmann category of foliations, introduced by the first author in [5, 9], is an invariant of foliated homotopy type with values in {1,2,...,1}. A foliation with all leaves compact and Hausdorff leaf space M/F is called compact Hausdorff. The transverse saturated category cat\| M of a compact Hausdorff foliation is always finite. In this paper we study the transverse category of compact Hausdorff foliations. Our main result provides upper and lower bounds on the transverse category cat\| (M) in terms of the geometry of F and the Epstein filtration of the exceptional set E. The exceptional set is the closed saturated foliated space which is the union of the leaves with non-trivial holonomy. We prove that max{cat(M/F),cat\| (E)} � cat\| (M) � cat\| (E) + q We give examples to show that both the upper and lower bounds are realized, so the estimate is sharp. We also construct a family of examples for which the transverse category for a compact Hausdorff foliation can be arbitrarily large, though the category of the leaf spaces is constant.

Jordan and alternative tori are the coordinate algebras of extended affine Lie algebras of types A 1 and A 2 . In this paper we show that the derivation algebra of a Jordan torus is a semidirect product of the ideal of inner derivations and the subalgebra of central derivations. In the course of proving this result, we investigate derivations of the more general class of division graded Jordan and alternative algebras. We also describe invariant forms of these algebras.

In this paper, we prove a fixed point formula for flat bundles. To this end, we use cyclic cocycles which are constructed out of closed invariant currents. We show that such cyclic cocycles are equivariant with respect to isometric longitudinal actions of compact Lie groups. This enables us to prove fixed point formulae in the cyclic homology of the smooth convolution algebra of the foliation.

In this paper we prove the smoothness of solutions of a class of elliptic-parabolic nonlinear Levi type equations, represented as a sum of squares plus a vector field. By means of a freezing method the study of the operator is reduced to the analysis of a family Lξ 0 of left invariant operators on a free nilpotent Lie group. The fundamental solution Γξ 0 of the operator Lξ 0 is used as a parametrix of the fundamental solution of the Levi operator, and provides an explicit representation formula for the solution of the given equation. Differentiating this formula and applying a bootstrap method, we prove that the solution is C∞.

This paper is part of a program to study the conjecture of E. C. Dade on counting characters in blocks for several finite groups. The invariant conjecture of Dade is proved for general linear and unitary groups when the characteristic of the modular representation is distinct from the defining characteristic of the groups.

In this paper we give a purely algebraic construction of the theory of residues of a Pfaff system relative to an invariant subscheme. This construction is valid over an arbitrary base scheme of any characteristic.