The most famous 2-dimensional continued fraction algorithm is the Jacobi algorithm. However, Brun and Selmer algorithms are also interesting 2-dimensional subtractive algorithms. Schratzberger shows that all these three algorithms are deeply related by a process similar to insertion and extension for continued fractions. In this note the basic ergodic properties of two mixtures of both maps are explored. Furthermore a digression to a quite different map is made which exhibits an “exotic” invariant measure.
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