The theorem on squaring a rectangle (see Schramm [6] and CannonFloyd-Parry [1]) gives a combinatorial version of the Riemann mapping theorem. We elucidate by example (the dumbbell) some of the limitations of rectangle-squaring as an approximation to the classical Riemamnn mapping.

We analyse dynamical properties of the negative beta transformation, which has been studied recently by Ito and Sadahiro. Contrary to the classical beta transformation, the density of the absolutely continuous invariant measure of the negative beta transformation may be zero on certain intervals. By investigating this property in detail, we prove that the (−β)-transformation is exact for all β ...

Abstract A group is said to have rational growth with respect a generating set if the series function. It was shown by Parry that certain torus bundle groups of even trace exhibits growth. We generalize this result class odd trace.

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