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In this chapter we prove that the number of elements of \(\mathcal {B}_3\diagup \mathcal {Z}_3\) with positive \(\varLambda _{tr}\) not exceeding a positive number Y  grows exponentially. As a corollary we give an alternative proof of the exponential growth of the number of conjugacy classes of elements of \(\mathcal {B}_3\diagup \mathcal {Z}_3\) that have positive entropy not exceeding Y . The corollary is a particular case of results of Veech and Eskin-Mirzakhani. Our proof does not use deep techniques from Teichmüller theory.

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Counting Functions

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  • 发表日期：

    2025年

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  • Burglind Jöricke

摘要

In this chapter we prove that the number of elements of \(\mathcal {B}_3\diagup \mathcal {Z}_3\) with positive \(\varLambda _{tr}\) not exceeding a positive number Y  grows exponentially. As a corollary we give an alternative proof of the exponential growth of the number of conjugacy classes of elements of \(\mathcal {B}_3\diagup \mathcal {Z}_3\) that have positive entropy not exceeding Y . The corollary is a particular case of results of Veech and Eskin-Mirzakhani. Our proof does not use deep techniques from Teichmüller theory.