Corpus ID: 236469552

# Boundary slopes for the Markov ordering on relatively prime pairs

@inproceedings{Gaster2021BoundarySF,
title={Boundary slopes for the Markov ordering on relatively prime pairs},
author={Jonah Gaster},
year={2021}
}
Following McShane, we employ the stable norm on the homology of the modular torus to investigate the Markov ordering on the set of relatively prime integer pairs (q, p) with q ≥ p ≥ 0. Our main theorem is a characterization of slopes along which the Markov ordering is monotone with respect to q, confirming conjectures of Lee-Li-Rabideau-Schiffler that refine conjectures of Aigner. The main tool is an explicit computation of the slopes at the corners of the stable norm ball for the modular torus… Expand

#### References

SHOWING 1-10 OF 41 REFERENCES
Convexity and Aigner's Conjectures
Markov numbers are integers that appear in triples which are solutions of a Diophantine equation, the so-called Markov cubic x + y + z − 3xyz = 0. A classical topic in number theory, these numbersExpand
On the ordering of the Markov numbers
• Mathematics
• 2020
The Markov numbers are the positive integers that appear in the solutions of the equation $x^2+y^2+z^2=3xyz$. These numbers are a classical subject in number theory and have important ramificationsExpand
Continued fractions and orderings on the Markov numbers
• Mathematics
• 2018
Markov numbers are integers that appear in the solution triples of the Diophantine equation, $x^2+y^2+z^2=3xyz$, called the Markov equation. A classical topic in number theory, these numbers areExpand
Continued Fractions in Cluster Algebras, Lattice Paths and Markov Numbers
In this work we present results from three different, albeit related, areas. First, we construct an explicit formula for the F-polynomial of a cluster variable in a surface type cluster algebra.Expand
On the Markov numbers: Fixed numerator, denominator, and sum conjectures
• Computer Science, Mathematics
• 2021
This paper generalizes Markov numbers to any couple (p,q) of nonnegative integers (not only when they are relatively primes) and conjecture that the unicity is still true as soon as $p \leq q$. Expand
Markov numbers, Mather’s β function and stable norm
• Mathematics
• Nonlinearity
• 2019
V. Fock [7] introduced an interesting function $\psi(x)$, $x \in {\mathbb R}$ related to Markov numbers. We explain its relation to Federer-Gromov's stable norm and Mather's $\beta$-function, and useExpand
Lifting Representations to Covering Groups
The primary purpose of this note is to prove that if a discrete subgroup Z of Z’S&(C) has no 2-torsion then it lifts to S&(C); i.e., there is a homomorphism Z+ S&(C) such that the composition withExpand
Lyapunov spectrum of Markov and Euclid trees
• Mathematics
• 2016
We study the Lyapunov exponents $\Lambda(x)$ for Markov dynamics as a function of path determined by $x\in \mathbb RP^1$ on a binary planar tree, describing the Markov triples and their "tropical"Expand
On the number of Markoff numbers below a given bound
According to a famous theorem of Markoff, the indefinite quadratic forms with exceptionally large minima (greater than f of the square root of the discriminant) are in 1 : 1 correspondence with theExpand
Mathematics of Computation
For each prime p, let p# be the product of the primes less than or equal to p. We have greatly extended the range for which the primality of n!± 1 and p#± 1 are known and have found two new primes ofExpand