The dynamics of Pythagorean Triples
Date
2022-04-27
Authors
Acar, Nazim
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Abstract
A Pythagorean Triple($PT$) is a triple of positive integers $(a,b,c)$, that satisfies\\
$a^2+b^2=c^2$. By requiring two of the entries being relatively prime, $(a,b,c)$ becomes a Primitive Pythagorean Triple($PPT$). This removes \emph{trivially equivalent} $PT$s.
Following up on the unpublished paper by D. Romik \cite{Romik} we develop a sequence of mappings and show how each $PPT$ has a unique path starting from one of the two initial nodes $(3,4,5),(4,3,5)$.\\
We explain a way of generating the $PPT$s through paper folding. \\
Using a various techniques from dynamics we show how these mappings can be carried over to their conjugates in the first unit arc $x^2+y^2=z^2, x,y\geq 0$ and the unit interval $[0,1]$. Under these mappings and through the conjugacies we show that the $PPT$s, the pair of rational points on the first unit arc and the rational numbers on the unit interval correspond to each other with the forward orbits exhibiting similar behavior.\\
We identify infinite, $\sigma$-finite invariant measures for one-dimensional systems. With the help of the developed conjugacies we extend the dynamics of the $PPT$s to the continued fraction expansion of the real numbers in the unit interval and show a connection to the Euclidean algorithm. We show that the dynamical system is conservative and ergodic.
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Keywords
PPT, Dynamics of Pythagorean Triples, Primitive Pythagorean Triples