Abstract
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.
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References
Ahlfors, L.A.: Lectures on Quasiconformal Maps. Princeton, NJ: van Nostrand Company, 1966
Herman, M.R.: Sur la conjugaison différentiable des difféomorphismes du cercle á des rotations. Publ. Math. I.H.E.S., No.49, 5–233 (1979)
Jiang, Y.: Generalized Ulam-von Neumann Transformations. Thesis (1990), Graduate Center of CUNY
Jiang, Y.: Geometry of geometrically finite one-dimensional maps. Commun. Math. Phys.156, 639–647 (1993)
Jiang, Y.: Local normalization of one dimensional maps. IHES preprint, June (1989)
Jiang, Y.: Asymptotic differentiable structure on Cantor set. Commun. Math. Phys.155, 503–509 (1993)
de la Llave, R., Moriyón, R.: Invariant for smooth conjugacy of hyperbolic dynamical system II. Commun. Math. Phys.109, 369–378 (1987)
Shub, M., Sullivan D.: Expanding endomorphisms of the circle revisited. Ergod. Th & Dynam. Sys.5, 285–289 (1987)
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Communicated by J.-P. Eckmann
The author is partially supported by a PSC-CUNY grant and a NSF grant.
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Jiang, Y. On Ulam-von Neumann transformations. Commun.Math. Phys. 172, 449–459 (1995). https://doi.org/10.1007/BF02101803
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DOI: https://doi.org/10.1007/BF02101803