On finite edge transitive graphs and rotary maps

It is shown that every connected vertex and edge transitive graph has a normal multicover that is a connected normal edge transitive Cayley graph. Moreover, every chiral or regular map has a normal cover that is a balanced chiral or regular Cayley map, respectively. As an application, a new family of half-transitive graphs is constructed as 2-fold covers of a family of 2-arc transitive graphs admitting Suzuki groups.     

圆周上逆极限可扩的连续自映射

本文研究了圆周上一类自映射ｆ的正向可扩性与其道极限的可扩性间的联系，得出圆周上的连续满射ｆ的逆极限可扩等价于ｆ拓扑共轭于扩张映射．     

Fixed-point-free maps on tree-like continua

A fixed-point-free homeomorphism on a tree-like continuum is described. The tree-like continuum is obtained as inverse limit of trees, and the homeomorphism is obtained as an induced map of the inverse limit space.     

Cremona maps defined by monomials

Cremona maps defined by monomials of degree 2 are thoroughly analyzed and classified via integer arithmetic and graph combinatorics. In particular, the structure of the inverse map to such a monomial Cremona map is made very explicit as is the degree of its monomial defining coordinates. As a special case, one proves that any monomial Cremona map of degree 2 has inverse of degree 2 if and only if it is an involution up to permutation in the source and in the target. This statement is subsumed in a recent result of L. Pirio and F. Russo, but the proof is entirely different and holds in all characteristics. One unveils a close relationship binding together the normality of a monomial ideal, monomial Cremona maps and Hilbert bases of polyhedral cones.     

Continuous maps on triangular matrices that preserve commutativity

We investigate continuous injective maps defined on the space of upper triangular matrices over a field F that preserve commutativity of matrices in both directions. We show that every such map is a composition of either an inner automorphism and a locally polynomial map or the two latter and one more automorphism of upper triangular matrices.     

Stable maps of Polish spaces

We define the notions of stable and transquotient maps and study the relation between these classes of maps. The class of stable maps contains all closed and open maps and their compositions. The transquotient maps preserve the property of being a Polish space, and every stable map between separable metric spaces is transquotient.

In particular, a composition of closed and open maps (the intermediary spaces may not be metric) preserves the property of being a Polish space. This generalizes the results of Sierpinski and Vainstein for open and closed maps.

     

Noncommutative ball maps

In this paper, we analyze problems involving matrix variables for which we use a noncommutative algebra setting. To be more specific, we use a class of functions (called NC analytic functions) defined by power series in noncommuting variables and evaluate these functions on sets of matrices of all dimensions; we call such situations dimension-free. These types of functions have recently been used in the study of dimension-free linear system engineering problems. In this paper we characterize NC analytic maps that send dimension-free matrix balls to dimension-free matrix balls and carry the boundary to the boundary; such maps we call “NC ball maps”. We find that up to normalization, an NC ball map is the direct sum of the identity map with an NC analytic map of the ball into the ball. That is, “NC ball maps” are very simple, in contrast to the classical result of D'Angelo on such analytic maps in C. Another mathematically natural class of maps carries a variant of the noncommutative distinguished boundary to the boundary, but on these our results are limited. We shall be interested in several types of noncommutative balls, conventional ones, but also balls defined by constraints called Linear Matrix Inequalities (LMI). What we do here is a small piece of the bigger puzzle of understanding how LMIs behave with respect to noncommutative change of variables.     

Transformations and non-degenerate maps

We shall prove the equivalences of a non-degenerate circle-preserving map and a Mobius transformation in Rn, of a non-degenerate geodesic-preserving map and an isometry in Hn, of a non-degenerate line-preserving map and an affine transformation in Rn. That a map is non-degenerate means that the image of the whole space under the map is not a circle, or geodesic or line respectively. These results hold without either injective or surjective, or even continuous assumptions, which are new and of a fundamental nature in geometry.     

Refinable and monotone maps revisited

Generalizing results by J. Ford, J. W. Rogers, Jr. and H. Kato we prove that (1) a map f from a G-like continuum onto a graph G is refinable iff f is monotone; (2) a graph G is an arc or a simple closed curve iff every G-like continuum that contains no nonboundary indecomposable subcontinuum admits a monotone map onto G.We prove that if bonding maps in the inverse sequence of compact spaces are refinable then the projections of the inverse limit onto factor spaces are refinable. We use this fact to show that refinable maps do not preserve completely regular or totally regular continua.     

Diffeomorphisms as time one maps

Summary. We investigate the inverse ODE problem of finding a vector field such that the time one map associated to its flow coincides with a given diffeomorphism. Using a constructive approach we solve this problem for a class of diffeomorphisms having a globally attracting fixed point. Furthermore we consider how the solution fields depend on the diffeomorphism. As an example we show that for certain parameters, the Hénon map is the time one map of a two dimensional flow.     

Counting unrooted maps on the plane

A planar map is a 2-cell embedding of a connected planar graph, loops and parallel edges allowed, on the sphere. A plane map is a planar map with a distinguished outside (“infinite”) face. An unrooted map is an equivalence class of maps under orientation-preserving homeomorphism, and a rooted map is a map with a distinguished oriented edge. Previously we obtained formulae for the number of unrooted planar n-edge maps of various classes, including all maps, non-separable maps, eulerian maps and loopless maps. In this article, using the same technique we obtain closed formulae for counting unrooted plane maps of all these classes and their duals. The corresponding formulae for rooted maps are known to be all sum-free; the formulae that we obtain for unrooted maps contain only a sum over the divisors of n. We count also unrooted two-vertex plane maps.     

Lifting factor maps to resolving maps

We consider Smale spaces, that is, homeomorphisms of a compact metric spaces possessing canonical coordinates of contracting (stable) and expanding (unstable) directions. Examples of such dynamical systems include the basic sets for Smale's Axiom A systems. We also assume that each point of the space is non-wandering and that there is a dense orbit. We show that any almost one-to-one factor map between two such systems may be lifted in a certain sense to a factor map which is injective on the local stable sets (i.e., s-resolving). We derive several corollaries. One is a refinement of Bowen's result that every irreducible Smale space is a factor of an irreducible shift of finite type by an almost one-to-one factor map. We are able to show that there exists such a factor which is the composition of an s-resolving map and a u-resolving map. Supported in part by a grant from NSERC, Canada.     

t-Balanced Cayley maps of dihedral groups

A Cayley map is an embedding of a Cayley graph where the cyclic ordering of generators around every vertex is the same. The involution indicating the position of mutually inverse generators in the cyclic ordering is called the distribution of inverses of a Cayley map. The Cayley maps whose distribution of inverses is linear (modulo the degree of the map) with 'slope' t are called t-balanced. An exponent of a Cayley map is a number e with the property that, roughly speaking, the Cayley map is isomorphic to its 'e-fold rotational image'.In the contribution we present results related to the construction of t-balanced Cayley maps which are not regular and do not have t as an exponent.     

On the Gray index conjecture for phantom maps

We study the Gray index, a numerical invariant for phantom maps. It has been conjectured that the only phantom map between finite-type spaces with infinite Gray index is the constant map. We disprove this conjecture by constructing a counter example. We also prove that this conjecture is valid if the target spaces of the phantom maps are restricted to being simply connected finite complexes.As a result of the counter example, we can show that SNT^∞(X) can be non-trivial for some space X of finite type.     

序列伪轨跟踪性与拓扑可迁

给出序列伪轨跟踪性的定义,得到拓扑可迁的一个充分条件,并证明,若f是同胚,则f具有序列伪轨跟踪性当且仅当其逆极限空间上的移位映射σf具有序列伪轨跟踪性。     

Tree-like continua and 2-to-1 maps

It is not known if there is a 2-to-1 map from a continuum onto a tree-like continuum. In fact, it is not known if there is a 2-to-1 map onto a hereditarily decomposable tree-like continuum. We show that the domain of such a map would have to contain an indecomposable continuum.

     

逆极限的不变测度和一致正熵性质

In this paper, the interconnection of some ergodic properties between a continuous selfmap and its inverse limit is studied. It has been proved that (1) their invariant Borel probability measures are identical up to homeomorphism and (2) they preserve uniform positive entropy property simuitaneously. As applications, it is also proved that the upper semi-continu-ous properties of their entropy maps are restricted each other, and the entropy map of the asymptotically h-expansive continuous map is upper semi-contlnuous, at the same time a continuous map having u, p.e. is topological weakmixing.     

F-actions and parallel-product decomposition of reflexible maps

The parallel product of two rooted maps was introduced by S.E. Wilson in 1994. The main question of this paper is whether for a given reflexible map M one can decompose the map into a parallel product of two reflexible maps. This can be achieved if and only if the monodromy (or the automorphism) group of the map has at least two minimal normal subgroups. All reflexible maps up to 100 edges, which are not parallel-product decomposable, are calculated and presented. For this purpose, all degenerate and slightly-degenerate reflexible maps are classified. In this paper the theory of F-actions is developed including a classification of quotients and parallel-product decomposition. Projections and lifts of automorphisms for quotients and for parallel products are studied. The theory can be immediately applied on rooted maps and rooted hypermaps as they are special cases of F-actions.     

Harmonic Riemannian maps on locally conformal Kaehler manifolds

We study harmonic Riemannian maps on locally conformal Kaehler manifolds (lcK manifolds). We show that if a Riemannian holomorphic map between lcK manifolds is harmonic, then the Lee vector field of the domain belongs to the kernel of the Riemannian map under a condition. When the domain is Kaehler, we prove that a Riemannian holomorphic map is harmonic if and only if the lcK manifold is Kaehler. Then we find similar results for Riemannian maps between lcK manifolds and Sasakian manifolds. Finally, we check the constancy of some maps between almost complex (or almost contact) manifolds and almost product manifolds.     

On existence of degenerate circle-preserving maps

Recently, B. Li and Y. Wang proved that if (n?2) is a circle-preserving map, then f is a Möbius transformation if and only if f is a non-degenerate map, where a map f is degenerate if the image is a circle. Furthermore, they conjectured that there should exist no degenerate map, or equivalently, f is a Möbius transformation if and only if f is a circle-preserving map. In this note, we construct a degenerate circle-preserving map to show that the conjecture is not true.