Homological identities for closed orbits

This paper deals with non-singular Smale flows on oriented 3-manifolds. We shall show a relation between the properties of invariant sets of a Smale flow and a kind of Whitehead torsion of the underlying manifold.

Non-singular Morse–Smale flows on 3-dimensional manifolds

We prove that the twisted De Rham cohomology of a flat vector bundleover some smooth manifold is isomorphic to the cohomology of invariant Pollicott–Ruelleresonant states associated with Anosov and Morse–Smale flows. As a consequence, weobtain generalized Morse inequalities for such flows. In the case of Morse–Smale flows,we relate the resonances lying on the imaginary axis with the twisted Fuller measuresused by Fried in his work on Reidemeister torsion. In particular, when V is a nonsingularMorse-Smale flow, we show that the Reidemeister torsion can be recovered from the onlyknowledge of dynamical resonances on the imaginary axis by expressing the torsion as azeta regularized infinite product of these resonances.

Haros graphs have been recently introduced as a set of graphs bijectively related to real numbers in the unit interval. Here we consider the iterated dynamics of a graph operator R over the set of Haros graphs. This operator was previously defined in the realm of graph-theoretical characterization of low-dimensional nonlinear dynamics and has a renormalization group (RG) structure. We find that the dynamics of R over Haros graphs is complex and includes unstable periodic orbits of arbitrary period and nonmixing aperiodic orbits, overall portraiting a chaotic RG flow. We identify a single RG stable fixed point whose basin of attraction is associated with the set of rational numbers, and find periodic RG orbits that relate to (pure) quadratic irrationals and aperiodic RG orbits, related with (nonmixing) families of nonquadratic algebraic irrationals and transcendental numbers. Finally, we show that the graph entropy of Haros graphs is globally decreasing as the RG flows towards its stable fixed point, albeit in a strictly nonmonotonic way, and that such graph entropy remains constant inside the periodic RG orbit associated to a subset of irrationals, the so-called metallic ratios. We discuss the possible physical interpretation of such chaotic RG flow and put results regarding entropy gradients along RG flow in the context of c-theorems.

The topological equivalence of low-dimensional Morse-Smale flows without fixed point (NMS-flows) under assumptions of various generality is the subject of a number of publications. Starting from dimension 4, there are only few results on classification. However, it is known that there exists nonsingular flows with wildly embedded invariant saddle manifolds. In this paper the class of nonsingular Morse-Smale flows on closed orientable 4-manifolds with a unique saddle orbit which is, moreover, nontwisted, is considered. It is shown that the equivalence class of a certain knot embedded in $\mathbb S^2\times\mathbb S^1$ is a complete invariant of such a flow. Given a knot in $\mathbb S^2\times\mathbb S^1$, a standard representative in the class of flows under consideration is constructed. The supporting manifold of all such flows is shown to be the manifold $\mathbb S^3\times\mathbb S^1$. Bibliography: 24 titles.

The set of periodic orbits of a non-singular Morse - Smale (NMS) flow on defines a link; a characterization of all possible links of NMS flows on has been developed by Wada. In the frame of codimension-one bifurcations, this characterization allows us to study the restrictions a link requires for suffering a given bifurcation. We also derive the topological description of the new link and the possibility of relating links by a chain of this type of bifurcation.

Surface-based computation of the Euler characteristic in the cubical grid

Any link in a 3-manifold is the closed orbits of a non-singular Morse-Smale flow after taking the split sum with the unlink and the connected sum with S2 × S1s.

Skew Weyl Modules for GLn and Degree Reduction for Schur Algebras

For large classes of non-convex subsets Y in $$\mathbb R^n$$ or in Riemannian manifolds (M, g) or in RCD-spaces (X, d, m) we prove that the gradient flow for the Boltzmann entropy on the restricted metric measure space $$(Y,d_Y,m_Y)$$ exists—despite the fact that the entropy is not semiconvex—and coincides with the heat flow on Y with Neumann boundary conditions.

A periodic point of a transformation of a set into itself is a point which is carried back to its original position by some iterate of the transformation. The purpose of this paper is to demonstrate topological conditions which ensure the existence of periodic points. THEOREM 1. Let T be a mapping of an orientable combinatorial manifold M into itself, and suppose that the corresponding homomorphism T* of rational p-cycle classes is an isomorphism onto. Let u be a rational p-cycle class and let v be any other rational cycle class of M. Let M be given a definite orientation. Then if the intersection cycle classes (T u, v), (T'u, v), , (T u, v) are all zero, where B is the p-dimensional Betti number of M, it follows that the intersection cycle class (u, v) is zero. PROOF. The rational p-cycle classes of M form a vector space of dimension B. Hence the B + 1 cycle classes u, T*u, ... , T*u must be related by a nontrivial dependency c0u + cjT*u + *. + CBT*U = 0. Let Ck, be the first nonzero coefficient. By hypothesis, T* has an inverse. Operating on the dependency by 71k gives CkU + Ck+lT*u + * + cBT* u = 0. Intersecting this combination of cycle classes with v and dividing by Ck gives the conclusion (u, v) = 0. THEOREM 2. Let T be a homeomorphism of an orientable combinatorial manifold M onto itself. If the Euler characteristic of M is not zero, then T has a periodic point. PROOF. Theorem 2 is a consequence of Theorem 1 applied to the orientable combinatorial product manifold M X M. Let M be given a definite orientation, thereby inducing an orientation of M X M. Let D be the cycle class in M X Ml which is the image of the basic cycle on M under the diagonal mapping x -* (x, x) of M into M X M. The homeomorphism T determines a homeomorphism 1 X T: (x, y) -+ (x, Ty) of M X M onto itself. If T has no periodic point, then the intersection classes ((1 X T)* D, D), ((1 X T) D, D), * a must all vanish. Hence by Theorem 1 the intersection class (D, D) must vanish. But a known result asserts that the algebraic value of the 0-cycle class (D, D) is equal within sign to the Euler characteristic of M. Since the Euler characteristic of M was assumed to be different from zero, T must have a periodic point. If Theorems 1 and 2 are established for manifolds with boundary then Theorem 2 can be extended to complexes by the imbedding method used by Lefschetz to extend his fixed-point formula. It is simpler, however, to make direct use of the Lefschetz formula. The method below was used by J. Nielsen in an investigation of transformations of surfaces. THEOREM 3. Let T be a mapping of a complex K into itself which induces auto-

We consider the optimal flows on noncompact surfaces with boundary, which have a minimum number of fixed points and all these points lie on the boundary of the surface. It is proved that the flow is optimal if it has a single sink and a single source. We describe the structures of the optimal flows on a simply connected region, on a Mobius strip, on a torus with hole, and on a Klein bottle with hole.