The paper constructs generalized discrete Markov spectra for the family of equations x² + y² + z² + k1 yz + k2 zx + k3 xy = (3 + k1 + k2 + k3) xyz, with each spectrum element realized as both a Lagrange constant of a quadratic irrational and a Markov constant of an indefinite binary quadratic form.
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Defines words ω_t via binary-tree recursion for each positive rational slope t; matrix evaluation of ω_t yields a Markov-monodromy matrix that encodes the generalized Markov number at t.
k-Markov numbers satisfy Aigner's conjectures on their orderings and uniqueness properties in positive integer solutions.
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Generalized discrete Markov spectra
The paper constructs generalized discrete Markov spectra for the family of equations x² + y² + z² + k1 yz + k2 zx + k3 xy = (3 + k1 + k2 + k3) xyz, with each spectrum element realized as both a Lagrange constant of a quadratic irrational and a Markov constant of an indefinite binary quadratic form.
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Words for generalized Markov numbers
Defines words ω_t via binary-tree recursion for each positive rational slope t; matrix evaluation of ω_t yields a Markov-monodromy matrix that encodes the generalized Markov number at t.
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Orderings of k-Markov Numbers
k-Markov numbers satisfy Aigner's conjectures on their orderings and uniqueness properties in positive integer solutions.