# The Arnol'd family of circle maps - origins and density of hyperbolicity


Arnol'd considers this family in the paper Small denominators I. Mapping the circle onto itself. The family is studied in Section 12 (in slightly different parametrization).

My first question is whether this is the first instance where the family has been suggested. On the Wikipedia page for this family, it is claimed that the family was introduced by Kolmogorov, but the editor in question has said that this is likely to have been a mistake. However, it is easy to believe that the family, which is very natural, could have been considered elsewhere independently, so I would be interested if anyone knows more about its origin. (I do not read Russian, but have asked a colleague to translate the relevant section in Arnol'd's paper, which does not seem to indicate a prior origin of the family.)

In a recent paper with van Strien, we were able to prove the density of hyperbolicity, i.e. of maps where both critical points belong to the basins of periodic attractors, in the non-invertible region $\mu_2>1/(2\pi)$. Density of hyperbolicity is essentially the central problem in one-dimensional dynamics (see Smale's 11th problem), and the Arnol'd family is one of the most-studied families of one-dimensional maps after (quadratic) polynomials. However, I am not aware of density of hyperbolicity in this family having been explicitly stated as an open problem in the literature before a recent paper of de Melo, Salomão and Vargas, which studied generalizations of this family with larger numbers of critical points. Again, does anyone know of this problem (or related ones, such as rigidity), being raised in the classical literature?

Many thanks for your help!

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