On Ulam-von Neumann transformations

We define and study Ulam-von Neumann transformations which are certain interval mappings and conjugate toq(x)=1–2x ² on –1,1]. We use a singular metric on –1,1] to study a Ulam-von Neumann transformation. This singular metric is universal in the sense that it does not depend on any particular mapping but only on the exponent of this mapping at its unique critical point. We give the smooth classification of Ulam-von Neumann transformations by their eigenvalues at periodic points and exponents and asymmetries.The author is partially supported by a PSC-CUNY grant and a NSF grant.