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:
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—Ralph Waldo Emerson (1803–1882)
“What joy when the insouciant
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What is this joy? That no animal
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“In my writing I am acting as a map maker, an explorer of psychic areas ... a cosmonaut of inner space, and I see no point in exploring areas that have already been thoroughly surveyed.”
—William Burroughs (b. 1914)