The Markov and Lagrange spectra on the Hecke group H4

초록

We consider the Markov spectrum and the Lagrange spectrum on the Hecke group H4. They are identical to the Markov and Lagrange spectra on the unit circle. The Markov spectrum on H4 is termed the Markov spectrum on index 2 sublattices by Vulakh and the Markov spectrum on 2-minimal forms or C-minimal forms by Schmidt. They characterized the spectrum up to the first accumulation point, independently. We show that, after the first accumulation point, both spectra have positive Hausdorff dimension. Then we find gaps in the spectra and give a bound on Hall's ray.

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