The Morse index,repeller-attractor pairs and the connection index for semiflows on noncompact spaces

This paper extends the Morse index theory of C. C. Conley to semiflows π on a noncompact meric space X. π is assumed to satisfy a hypothesis related to conditional α-contraction. We collect background material, define quasi-index pairs and the Morse index of a compact, isolated invariant set K, and prove that the Morse index is a connected simple system. We study repeller-attractor pairs in K, define index triples, and prove their existence and several properties leading to the concepts of the connection index, the connection map and the splitting class. Finally, we consider paths (continuous families) of pairs (π, K) and study continuations of the Morse and the connection indices along such paths. The present paper is a sequel to the author's previous work: On the homotopy index for infinite-dimensional semiflows (Trans. Amer. Math. Soc.269 (1982), 351–382).     

G-invariante Morse-funktionen

If f is a Morse function on a smooth manifold M there exists a homotopy equivalence from M to a CW complex X such that the critical points of f with index are in a one-one correspondence to the -cells of X. In the equivariant case, a similar result holds for a special type of invariant Morse functions. In this paper we prove the existence of such special invariant Morse functions on compact smooth G-manifolds. As a consequence, any compact smooth G-manifold is homotopy equivalent to a G-CW complex. Other applications deal with the Euler number of the fixed point set and Morse inequalities in equivariant homology theory.     

Morse Decompositions with Infinite Components for Multivalued Semiflows

In this paper we study the theory of Morse decompositions with an infinite number of components in the multivalued framework, proving that for a disjoint infinite family of weakly invariant sets (being all isolated but one) a Lyapunov function ordering them exists if and only if the multivalued semiflow is dynamically gradient. Moreover, these properties are equivalent to the existence of a Morse decomposition.This theorem is applied to a reaction-diffusion inclusion with an infinite number of equilibria.     

All-time Morse decompositions of linear nonautonomous dynamical systems

Morse decompositions provide inside information about the global asymptotic behavior of dynamical systems on compact metric spaces. Recently, the existence of Morse decompositions for nonautonomous dynamical systems was proved by restricting attention to the past or the future of the system, but in general, such a construction is not realizable for the entire time. In this article, it is shown that all-time Morse decompositions can be defined for linear systems on the projective space. Moreover, the dynamical properties are discussed and an analogue to the Theorem of Selgrade is proved.

     

Strongly invertible links and divides

To a proper generic immersion of a finite number of copies of the unit interval in a 2-disc, called a divide, A’Campo associates a link in S³. From the more general notion of ordered Morse signed divides, one obtains a braid presentation of links of divides. In this paper, we prove that every strongly invertible link is isotopic to the link of an ordered Morse signed divide. We give fundamental moves for ordered Morse signed divides and show that strongly invertible links are equivalent if and only if we can pass from one ordered Morse signed divide to the other by a sequence of such moves. Then we associate a polynomial to an ordered Morse signed divide, invariant for these moves. So this polynomial is invariant for the equivalence of strongly invertible links.     

Representation dimension: An invariant under stable equivalence

In this paper, we prove that the representation dimension is an invariant under stable equivalence.

     

Conley–Morse–Forman Theory for Combinatorial Multivector Fields on Lefschetz Complexes

We introduce combinatorial multivector fields, associate with them multivalued dynamics and study their topological features. Our combinatorial multivector fields generalize combinatorial vector fields of Forman. We define isolated invariant sets, Conley index, attractors, repellers and Morse decompositions. We provide a topological characterization of attractors and repellers and prove Morse inequalities. The generalization aims at algorithmic analysis of dynamical systems through combinatorialization of flows given by differential equations and through sampling dynamics in physical and numerical experiments. We provide a prototype algorithm for such applications.     

Chaotic sets of continuous and discontinuous maps

This paper discusses Li-Yorke chaotic sets of continuous and discontinuous maps with particular emphasis to shift and subshift maps. Scrambled sets and maximal scrambled sets are introduced to characterize Li-Yorke chaotic sets. The orbit invariant for a scrambled set is discussed. Some properties about maximality, equivalence and uniqueness of maximal scrambled sets are also discussed. It is shown that for shift maps the set of all scrambled pairs has full measure and chaotic sets of some discontinuous maps, such as the Gauss map, interval exchange transformations, and a class of planar piecewise isometries, are studied. Finally, some open problems on scrambled sets are listed and remarked.     

Morse Decompositions for Periodic General Dynamical Systems and Differential Inclusions

We first establish the Morse decomposition theory of periodic invariant sets for non-autonomous periodic general dynamical systems (set-valued dynamical systems). Then we discuss the stability of Morse decompositions of periodic uniform forward attractors. We also apply the abstract results to non-autonomous periodic differential inclusions with only upper semi-continuous right-hand side. We show that Morse decompositions are robust with respect to both internal and external perturbations (upper semi-continuity of Morse sets). Finally as an application we study the effect of small time delays to asymptotic behavior of control systems from the point of view of Morse decompositions.     

Morse decompositions of nonautonomous dynamical systems

The global asymptotic behavior of dynamical systems on compact metric spaces can be described via Morse decompositions. Their components, the so-called Morse sets, are obtained as intersections of attractors and repellers of the system. In this paper, new notions of attractor and repeller for nonautonomous dynamical systems are introduced which are designed to establish nonautonomous generalizations of the Morse decomposition. The dynamical properties of these decompositions are discussed, and nonautonomous Lyapunov functions which are constant on the Morse sets are constructed explicitly. Moreover, Morse decompositions of one-dimensional and linear systems are studied.

     

Multiple solutions for elliptic equations at resonance

We study an asymptotically linear elliptic equation at resonance, with an odd nonlinearity. By a penalization technique and suitable min-max theorems (which give Morse index estimates), we prove the existence of pairs of non trivial solutions, where N is, roughly speaking, the difference between the Morse indexes at zero and at infinity. Received December 1999     

Dual decompositions of 4-manifolds II: Linear invariants

This paper continues the study of decompositions of a smooth 4-manifold into two handlebodies with handles of index . Part I (Trans. Amer. Math. Soc. 354 (2002), 1373-1392) gave existence results in terms of spines and chain complexes over the fundamental group of the ambient manifold. Here we assume that one side of a decomposition has larger fundamental group, and use this to define algebraic-topological invariants. These reveal a basic asymmetry in these decompositions: subtle changes on one side can force algebraic-topologically detectable changes on the other. A solvable iteration of the basic invariant gives an ``obstruction theory' using lower commutator quotients. By thinking of a 2-handlebody as essentially determined by the links used as attaching maps for its 2-handles, this theory can be thought of as giving ``ambient' link invariants. The moves used are related to the grope cobordism of links developed by Conant-Teichner, and the Cochran-Orr-Teichner filtration of the link concordance groups. The invariants give algebraically sophisticated ``finite type' invariants in the sense of Vassilaev.

     

On the structure of invariant attracting sets of systems of finite-difference equations generating Hénon-Lozi mappings

For Hénon-Lozi mappings F, we find sufficient conditions under which on the plane there exists a domain U such that its closure is mapped by F strictly inside U. This ensures the existence of a compact invariant set in U. We prove the existence of an open set of parameter values for which this invariant set contains a zero-dimensional locally maximal topologically transitive Markov set such that the restriction of the mapping to this set is topologically conjugate to the shift automorphism in the space of sequences of two symbols. We show that if this Markov set is hyperbolic, then the above-mentioned compact invariant set coincides with the closure of the unstable manifold of F at a fixed point lying in that set and is a topologically indecomposable one-dimensional continuum. We present the parameter values for which these results hold for the Hénon mapping. We thereby prove the existence of a parameter range in which the invariant set of the Hénon mapping is a one-dimensional topologically indecomposable Brauer-Janiszewski continuum that contains a zero-dimensional locally maximal set and lies in the attraction domain of itself.     

Discrete morse theory and the cohomology ring

In [5], we presented a discrete Morse Theory that can be applied to general cell complexes. In particular, we defined the notion of a discrete Morse function, along with its associated set of critical cells. We also constructed a discrete Morse cocomplex, built from the critical cells and the gradient paths between them, which has the same cohomology as the underlying cell complex. In this paper we show how various cohomological operations are induced by maps between Morse cocomplexes. For example, given three discrete Morse functions, we construct a map from the tensor product of the first two Morse cocomplexes to the third Morse cocomplex which induces the cup product on cohomology. All maps are constructed by counting certain configurations of gradient paths. This work is closely related to the corresponding formulas in the smooth category as presented by Betz and Cohen [2] and Fukaya [11], [12].

     

On measures ergodic with respect to an analytic equivalence relation

In this paper, we prove that the set of probability measures which are ergodic with respect to an analytic equivalence relation is an analytic set. This is obtained by approximating analytic equivalence relations by measures, and is used to give an elementary proof of an ergodic decomposition theorem of Kechris.

     

The existence and uniqueness of trajectories joining critical points for differential equations in R3

In this paper, we prove the existence and uniqueness of trajectories joining critical points for differential equations in R³ by constructing the index pair of the isolated invariant set and using Conley index theory.     

Conformal metrics with prescribed Q-curvature on S n

In this paper we prescribe a fourth order conformal invariant on the standard n-sphere, with n????5, and study the related fourth order elliptic equation. We prove new existence results based on a new type of Euler?CHopf type formula. Our argument gives an upper bound on the Morse index of the obtained solution. We also give a lower bound on the number of conformal metrics having the same Q-curvature.     

Unstable manifold,Conley index and fixed points of flows

We study dynamical and topological properties of the unstable manifold of isolated invariant compacta of flows. We show that some parts of the unstable manifold admit sections carrying a considerable amount of information. These sections enable the construction of parallelizable structures which facilitate the study of the flow. From this fact, many nice consequences are derived, specially in the case of plane continua. For instance, we give an easy method of calculation of the Conley index provided we have some knowledge of the unstable manifold and, as a consequence, a relation between the Brouwer degree and the unstable manifold is established for smooth vector fields. We study the dynamics of non-saddle sets, properties of existence or non-existence of fixed points of flows and conditions under which attractors are fixed points, Morse decompositions, preservation of topological properties by continuation and classify the bifurcations taking place at a critical point.     

Twist points of the von Koch snowflake

It is known that the set of twist points in the boundary of the von Koch snowflake domain has full harmonic measure. We provide a new, simple proof, based on the doubling property of the harmonic measure, and on the existence of an equivalent measure, invariant and ergodic with respect to the shift.

     

Graded quaternion symbol equivalence of function fields

We present criteria for a pair of maps to constitute a quaternion-symbol equivalence (or a Hilbert-symbol equivalence if we deal with global function fields) expressed in terms of vanishing of the Clifford invariant. In principle, we prove that a local condition of a quaternion-symbol equivalence can be transcribed from the Brauer group to the Brauer-Wall group.