Abstract
The concepts of train track was introduced by W. P. Thurston to study the measured foliations/laminations and the pseudo-Anosov mapping classes on a surface. In this talk, we translate some concepts of train tracks into the language of cluster algebras using the tropicalization of Goncharov{Shen’s potential function. Using this, we translate a combinatorial property of a train track associated with a pseudo-Anosov mapping class into the combinatorial property in cluster algebras, called the sign stability which was introduced by Tsukasa Ishibashi and the speaker.
Definitions
We define Foliations of marked surfaces. We then define the transverse measure. We define the space of measured foliations. And a whitehead move.
There is an embedding from the space of Train tracks into the space of measure foliations. Especially we can get an atlas of the foliation space.
More definition
Decorated measured foliations allow special structures around punctures.
Potential function
Goncharov-Shen introduce a landau-Ginzburg potential function on a cluster -varieties. This gives a weight to boundary components.
More definitions
Splitting and shifting is defined.
About Sign stability
Ishibashi-K. defined sign stability as a combinatorial property for mutation loops (generalization of mapping classes) as analogy of the pseudo-Anosov class.
Calculation of entropies
We can use this to calculate algebraical and categorical entropy