scholarly journals Asymptotic flag of an orientable measured foliation

1993 ◽  
Vol 11 ◽  
pp. 113-131
Author(s):  
Anton Zorich
Keyword(s):  
Measured Foliation

scholarly journals MASS GROWTH OF OBJECTS AND CATEGORICAL ENTROPY

2020 ◽  
pp. 1-22
Author(s):  
AKISHI IKEDA
Keyword(s):  
Classical Theory ◽  
Stability Condition ◽  
Triangulated Category ◽  
Categorical Theory ◽  
Mass Growth ◽  
Fundamental Properties ◽  
Measured Foliation ◽  
The Stability ◽  
The Relationship

In the pioneering work by Dimitrov–Haiden–Katzarkov–Kontsevich, they introduced various categorical analogies from the classical theory of dynamical systems. In particular, they defined the entropy of an endofunctor on a triangulated category with a split generator. In the connection between the categorical theory and the classical theory, a stability condition on a triangulated category plays the role of a measured foliation so that one can measure the “volume” of objects, called the mass, via the stability condition. The aim of this paper is to establish fundamental properties of the growth rate of mass of objects under the mapping by the endofunctor and to clarify the relationship between it and the entropy. We also show that they coincide under a certain condition.

Thurston's Proof

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
General Theory ◽  
Linear Algebra ◽  
Point Theorem ◽  
Original Proof ◽  
Closed Curves ◽  
Measured Foliation ◽  
Geodesic Laminations ◽  
Train Tracks

This chapter describes Thurston's original path of discovery to the Nielsen–Thurston classification theorem. It first provides an example that illustrates much of the general theory, focusing on Thurston's iteration of homeomorphisms on simple closed curves as well as the linear algebra of train tracks. It then explains how the general theory works and presents Thurston's original proof of the Nielsen–Thurston classification. In particular, it considers the Teichmüller space and the measured foliation space. The chapter also discusses measured foliations on a pair of pants, global coordinates for measured foliation space, the Brouwer fixed point theorem, the Thurston compactification for the torus, and Markov partitions. Finally, it evaluates other approaches to proving the Nielsen–Thurston classification, including the use of geodesic laminations.

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