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.
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Famous quotes containing the words train, track and/or map:
“The train was now going fast. Franz suddenly clutched his side, transfixed by the thought that he had lost his wallet which contained so much.”
—Vladimir Nabokov (1899–1977)
“It is remarkable how easily and insensibly we fall into a particular route, and make a beaten track for ourselves. I had not lived there a week before my feet wore a path from my door to the pond-side; and though it is five or six years since I trod it, it is still quite distinct. It is true, I fear, that others may have fallen into it, and so helped to keep it open.”
—Henry David Thoreau (1817–1862)
“When I had mapped the pond ... I laid a rule on the map lengthwise, and then breadthwise, and found, to my surprise, that the line of greatest length intersected the line of greatest breadth exactly at the point of greatest depth.”
—Henry David Thoreau (1817–1862)