Combinatorics of Train Tracks. (AM-125)

Book Review: Teichmüller theory and applications to geometry, topology, and dynamics. Volume 1: Teichmüller theory; Teichmüller theory and applications to geometry, topology, and dynamics. Volume 2: Surface homeomorphisms and rational functions

“…As a matter of fact, Levy's thesis is probably the first place where Thurston's theorem on the characterization of rational maps was used. To give another example, Penner's name is completely absent from the bibliography of this volume, whereas it should be there, at least for his lucid exposition, with Harer, of train tracks [32], for his general construction of pseudo-Anosov diffeomorphisms [31], his algorithms to find them, and for his work on the dilatation factors of these maps, a topic which is discussed in this book.…”

Confluent Drawings: Visualizing Non-planar Diagrams in a Planar Way

“…Although we are not familiar with any prior work on the automatic display of graphs using this confluent diagram approach, we have observed that some airlines use hand-crafted confluent diagrams to display their route maps. Diagrams similar to our confluent drawings have also been used by Penner and Harer [33] to study the topology of surfaces. In addition to providing heuristic algorithms for recognizing and drawing confluent diagrams, we also show that there are large classes of non-planar graphs that can be drawn in a planar way using our confluent diagram approach.…”

Approximate roots of pseudo-Anosov diffeomorphisms

“…We recall here facts about train tracks and bigon tracks, see [20], [18]. Let τ ⊂ Z be a smooth 1-dimensional branched manifold: thus τ is a 1-dimensional CW-complex in which the interiors of edges are smooth curves, and the field of TOME 59 (2009), FASCICULE 4 tangent lines T x τ, x ∈ τ \ {vertices}, extends to a continuous line field on τ.…”

“…Note also that the carrying matrix of the inverse Φ −1 is M T Φ . See [18] for more discussion of this. When Φ is a pseudo Anosov of Thurston-Penner type, we may calculate the carrying matrix using the bigon track defined in §4.…”