$q$-Deformations and $t$-deformations of Markov triples

An equivalent condition for the Markov triples and the Diophantine equation $a^2+b^2+c^2=abcf(a,b,c)$

We propose an equivalent condition for the Markov triples, which was mentioned by H. Rademacher essentially. As an application, we study the solvability of the Diophantine equation $a^2+b^2+c^2=abcf(a,b,c)$.

Lyapunov spectrum of Markov and Euclid trees

We study the Lyapunov exponents $Λ(x)$ 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" version - Euclid triples. We show that the corresponding Lyapunov spectrum is $[0, \ln φ]$, where $φ$ is the golden ratio, and prove that on the Markov-Hurwitz set $\mathbb{X}$ of the most irrational numbers the corresponding function $Λ_\mathbb{X}$ is monotonically increasing and in the Farey parametrization is convex.

k-马尔科夫数的序

Orderings of k-Markov Numbers

本文研究k-马尔科夫数的序问题，证明其满足Aigner的猜想，基于簇代数理论。

Continued fractions and orderings on the Markov numbers

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 are related to many areas of mathematics such as combinatorics, hyperbolic geometry, approximation theory and cluster algebras. There is a natural map from the rational numbers between zero and one to the Markov numbers. In this paper, we prove two conjectures seen in Martin Aigner's book, Markov's theorem and 100 years of the uniqueness conjecture, that determine an ordering on subsets of the Markov numbe…

Uniqueness conjectures for extended Markov numbers

We study an extension to the uniqueness conjecture for Markov numbers. For any three positive integers $m\geq a$ and $m\geq b$ satisfying $a^2+b^2+m^2=3abm$, this conjecture states that the triple $(a,m,b)$ is uniquely determined by the Markov number $m$. The theory of Markov numbers may be described by combinatorics of the sequences $(1,1)$ and $(2,2)$. There is an extension to the theory based on arbitrary sequences. We define extended uniqueness conjectures for any sequences $μ$ and $ν$. We show that for certain integers $a>1$ and $b>2$ the extended uniqueness conjecture for the sequences $…

素数幂形式的广义马尔可夫数的唯一性定理

Uniqueness theorem of generalized Markov numbers that are prime powers

该研究将经典马尔可夫方程推广为带参数k的广义形式，拓展Cohn三元组概念，证明了素数幂形式的广义马尔可夫数的唯一性定理。