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We study the values of the recently introduced involution J (jimm) of thereal line, which is equivariant with the action of the group PGL(2,Z). We testour conjecture that this involution sends algebraic numbers of degree at leastthree to transcendental values. We also deduce some theoretical resultsconcerning the continued fraction statistics of the generic values of thisinvolution and compare them with the experimental results.

Testing the transcendence conjectures of a modular involution of the real line and its continued fraction statistics