Least number of n-periodic points of self-maps of $$PSU(2)\times PSU(2)$$

Let $$f:M\rightarrow M$$ f : M → M be a self-map of a compact manifold and $$n\in {\mathbb {N}}$$ n ∈ N . In general, the least number of n-periodic points in the smooth homotopy class of f may be much bigger than in the continuous homotopy class. For a class of spaces, including compact Lie groups, a necessary condition for the equality of the above two numbers, for each iteration $$f^n$$ f n , appears. Here we give the explicit form of the graph of orbits of Reidemeister classes $$\mathcal {GOR}(f^*)$$ GOR ( f ∗ ) for self-maps of projective unitary group PSU(2) and of $$PSU(2)\times PSU(2)$$ P S U ( 2 ) × P S U ( 2 ) satisfying the necessary condition. The structure of the graphs implies that for self-maps of the above spaces the necessary condition is also sufficient for the smooth minimal realization of n-periodic points for all iterations.

Realization of homotopy classes of torus homeomorphisms by the simplest structurally stable diffeomorphisms

According to Thurston’s classification, the set of homotopy classes of orientation-preserving homeomorphisms of orientable surfaces is split into four disjoint subsets. A homotopy class from each subset is characterized by the existence of a homeomorphism called Thurston’s canonical form, namely: a periodic homeomorphism, a reducible nonperiodic homeomorphism of algebraically finite order, a reducible homeomorphism that is not a homeomorphism of an algebraically finite order, and a pseudo-Anosov homeomorphism. Thurston’s canonical forms are not structurally stable diffeomorphisms. Therefore, the problem naturally arises of constructing the simplest (in a certain sense) structurally stable diffeomorphisms in each homotopy class. In this paper, the problem posed is solved for torus homeomorphisms. In each homotopy class, structurally stable representatives are analytically constructed, namely, a gradient-like diffeomorphism, a Morse-Smale diffeomorphism with an orientable heteroclinic, and an Anosov diffeomorphism, which is a particular case of a pseudo-Anosov diffeomorphism.

Geometric Approach to Optimal Path Problem with Uncertain Arc Lengths

In this paper, the problem of finding the shortest paths, one of the most important problems in science and technology has been geometrically studied. Shortest path algorithm has been generalized to the shortest cycles in each homotopy class on a surface with arbitrary topology, using the universal covering space notion in the algebraic topology. Then, a general algorithm has been presented to compute the shortest cycles (geometrically rather than combinatorial) in each homotopy class. The algorithm can handle surface meshes with the desired topology, with or without boundary. It also provides a fundamental framework for other algorithms based on universal coverage space due to the capacity and flexibility of the framework.

Multistring based matrices

A virtual[Formula: see text]-string is a chord diagram with [Formula: see text] core circles and a collection of arrows between core circles. We consider virtual [Formula: see text]-strings up to virtual homotopy, compositions of flat virtual Reidemeister moves on chord diagrams. Given a virtual 1-string [Formula: see text], Turaev associated a based matrix that encodes invariants of the virtual homotopy class of [Formula: see text]. We generalize Turaev’s method to associate a multistring based matrix to a virtual [Formula: see text]-string, addressing an open problem of Turaev and constructing similar invariants for virtual homotopy classes of virtual [Formula: see text]-strings.

Minimizing intersection points of curves under virtual homotopy

A flat virtual link is a finite collection of oriented closed curves [Formula: see text] on an oriented surface [Formula: see text] considered up to virtual homotopy, i.e., a composition of elementary stabilizations, destabilizations, and homotopies. Specializing to a pair of curves [Formula: see text], we show that the minimal number of intersection points of curves in the virtual homotopy class of [Formula: see text] equals to the number of terms of a generalization of the Anderson–Mattes–Reshetikhin Poisson bracket. Furthermore, considering a single curve, we show that the minimal number of self-intersections of a curve in its virtual homotopy class can be counted by a generalization of the Cahn cobracket.