In the mathematical subject of geometric group theory a train track map is a continuous map f from a finite connected graph to itself which is a homotopy equivalence and which has particularly nice cancellation properties with respect to iterations. This map sends vertices to vertices and edges to nontrivial edge-paths with the property that for every edge e of the graph and for every positive integer n the path fn(e) is immersed, that is fn(e) is locally injective on e. Train-track maps are a key tool in analyzing the dynamics of automorphisms of finitely generated free groups and in the study of the Culler–Vogtmann Outer space.
Read more about Train Track Map: History, Examples, Applications and Generalizations, See Also, Basic References
Famous quotes containing the words train, track and/or map:
“Constant revolutionizing of production ... distinguish the bourgeois epoch from all earlier ones. All fixed, fast-frozen relations, with their train of ancient and venerable prejudices are swept away, all new-formed ones become antiquated before they can ossify. All that is solid melts into air, all that is holy is profaned, and man is at last compelled to face with sober senses, his real conditions of life, and his relations with his kind.”
—Karl Marx (1818–1883)
“The universe constantly and obediently answers to our conceptions; whether we travel fast or slow, the track is laid for us. Let us spend our lives in conceiving then. The poet or the artist never yet had so fair and noble a design but some of his posterity at least could accomplish it.”
—Henry David Thoreau (1817–1862)
“The Management Area of Cherokee
National Forest, interested in fish,
Has mapped Tellico and Bald Rivers
And North River, with the tributaries
Brookshire Branch and Sugar Cove Creed:
A fishy map for facile fishery....”
—Allen Tate (1899–1979)