arXiv subjects

K. Spalding

Publications and source records attributed to K. Spalding.

4 recordsLinked to original sources

arXiv2020

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.

math.DS↗

arXiv2020

Growth of values of binary quadratic forms and Conway rivers

We study the growth of the values of binary quadratic forms $Q$ on a binary planar tree as it was described by Conway. We show that the corresponding Lyapunov exponents $Λ_Q(x)$ as a function of the path determined by $x\in \mathbb RP^1$ are twice the values of the corresponding exponents for the growth of Markov numbers \cite{SV}, except for the paths corresponding to the Conway rivers, when $Λ_Q(x)=0.$ The relation with Galois results about continued fraction expansions for quadratic irrationals is explained and interpreted geometrically.

math.DS↗

arXiv2019

Tropical Markov dynamics and Cayley cubic

We study the tropical version of Markov dynamics on the Cayley cubic, introduced by V.E. Adler and one of the authors. We show that this action is semi-conjugated to the standard action of $SL_2(\mathbb Z)$ on a torus, and thus is ergodic with the Lyapunov exponent and entropy given by the logarithm of the spectral radius of the corresponding matrix.

math.DS↗

arXiv2018

Conway river and Arnold sail

We establish a simple relation between two geometric constructions in number theory: the Conway river of a real indefinite binary quadratic form and the Arnold sail of the corresponding pair of lines.

math.NT↗