A characterization of pseudo-Anosov mapping classes via tropical cluster transformations

Abstract

We give a characterization of generic pseudo-Anosov mapping classes purely in terms of their expressions in the shear coordinates, thus giving an answer to a problem raised by Papadopoulos--Penner [PP93]. This characterization has a cluster algebraic generalization called the sign stability introduced in [IK21]. By combining with the results in [IK21], we see that the algebraic entropies of the cluster $\mathcal{A}$- and $\mathcal{X}$-transformations induced by a generic pseudo-Anosov mapping class both coincide with its topological entropy.

Previous title is "Sign stability of mapping classes on marked surfaces I: empty boundary case". Compare to the previous paper, the review part is simplified and the characterization is concentrated on the "generic" pA

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Cited by in corpus (3)

• Earthquake theorem for cluster algebras of finite type
• Sign stability of mapping classes on marked surfaces II: general case via reductions
• Categorical dynamical systems arising from sign-stable mutation loops