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

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