M ay 2 00 8 Rotation set for maps of degree 1 on the graph sigma
نویسنده
چکیده
For a continuous map on a topological graph containing a unique loop S it is possible to define the degree and, for a map of degree 1, rotation numbers. It is known that the set of rotation numbers of points in S is a compact interval and for every rational r in this interval there exists a periodic point of rotation number r. The whole rotation set (i.e. the set of all rotation numbers) may not be connected and it is not known in general whether it is closed. The graph sigma is the space consisting in an interval attached by one of its endpoints to a circle. We show that, for a map of degree 1 on the graph sigma, the rotation set is closed and has finitely many connected components. Moreover, for all rational numbers r in the rotation set, there exists a periodic point of rotation number r.
منابع مشابه
ar X iv : m at h - ph / 0 30 50 57 v 1 2 7 M ay 2 00 3 Rhombic embeddings of planar graphs with faces of degree 4 Richard
Given a finite or infinite planar graph all of whose faces have degree 4, we study embeddings in the plane in which all edges have length 1, that is, in which every face is a rhombus. We give a necessary and sufficient condition for the existence of such an embedding, as well as a description of the set of all such embeddings.
متن کاملar X iv : m at h - ph / 0 30 50 57 v 1 2 7 M ay 2 00 3 Rhombic embeddings of planar graphs with faces of degree 4
Given a finite or infinite planar graph all of whose faces have degree 4, we study embeddings in the plane in which all edges have length 1, that is, in which every face is a rhombus. We give a necessary and sufficient condition for the existence of such an embedding, as well as a description of the set of all such embeddings.
متن کامل1 2 M ay 2 00 6 Bounds on graph eigenvalues
We refute, improve or amplify some recent results on graph eigenvalues. In particular, we prove that if G is a graph of order n ≥ 2, maximum degree ∆, and girth at least 5, then the maximum eigenvalue μ (G) of the adjacency matrix of G satisfies μ (G) ≤ min {
متن کامل1 8 M ay 2 00 8 COPS AND ROBBERS IN A RANDOM GRAPH DRAFT BÉLA
We consider the pursuit and evasion game on finite, connected, undirected graphs known as cops and robbers. Meyniel conjectured that for every graph on n vertices O(n 1
متن کاملar X iv : m at h / 02 05 33 5 v 1 [ m at h . Q A ] 3 1 M ay 2 00 2 Yang - Baxter maps and integrable dynamics
The hierarchy of commuting maps related to a set-theoretical solution of the quantum Yang-Baxter equation (Yang-Baxter map) is introduced. They can be considered as dynamical analogues of the monodromy and transfer-matrices. The general scheme of producing Yang-Baxter maps based on matrix factorisation is described. Some examples of birational Yang-Baxter maps appeared in the KdV theory are dis...
متن کامل