Abstract
In this survey, I summarize some work towards understanding of differential rigidity and smooth conjugacy in one-dimensional dynamics. In particular, I focus on those dynamical systems that have critical points and on those dynamical systems that have only \({C}^{1+\alpha },0 < \alpha < 1\), smoothness.
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References
Ferreira, F., Pinto, A.A.: Explosion of smoothness from a point to everywhere for conjugacies between diffeomorphisms on surfaces. Ergod. Theory Dynam. Syst. 23, 509–517 (2003)
Jiang, Y.: Teichmüller structures and dual geometric Gibbs type measure theory for continuous potentials. Preprint.
Jiang, Y.: Function model of the Teichmüller space of a closed hyperbolic Riemann surface. Preprint.
Jiang, Y.: Generalized Ulam-von Neumann transformations. Ph.D. Thesis, Graduate School of CUNY and UMI publication (1990)
Jiang, Y.: Dynamics of Certain Smooth One-Dimensional Mappings. I: The C 1 + α-Denjoy-Koebe Distortion Lemma; II. Geometrically Finite One-Dimensional Mappings; III. Scaling Function Geometry; IV. Asymptotic Geometry of Cantor Sets. IMS preprint series, 91-1a, 91-1b, 91-12a, 91-12b
Jiang, Y.: Differentiable rigidity and smooth conjugacy. Ann. Acad. Sci. Fenn. Math. 30, 361–383 (2005)
Jiang, Y.: Smooth classification of geometrically finite one-dimensional maps. Trans. Am. Math. Soc., 348(6), 2391–2412 (1996)
Jiang, Y.: On rigidity of one-dimensional maps. Contemp. Math., AMS Series, 211, 319–431 (1997)
Jiang, Y.: On Ulam-von Neumann transformations. Comm. Math. Phys. 172(3), 449–459 (1995)
Jiang, Y.: Geometry of geometrically finite one-dimensional maps. Comm. Math. Phys. 156(3), 639–647 (1993)
Jiang, Y.: Renormalization and Geometry in One-Dimensional and Complex Dynamics. Advanced Series in Nonlinear Dynamics, vol. 10, xvi + 309 pp. World Scientific Publishing Co. Pte. Ltd., River Edge, NJ, ISBN 981-02-2326-9 (1996)
de Melo, W., van Strien, S.: One-Dimensional Dynamics. Springer, Berlin (1993)
Mostow, D.: Strong rigidity of locally symmetric spaces. Ann. of Math. Stud., vol. 78. Princeton, NJ (1972)
Shub, M.: Endomorphisms of compact differentiable manifolds. Am. J. Math. 91, 129–155 (1969)
Shub, M., Sullivan, D.: Expanding Endomorphisms of the circle revisited. Ergod. Theroy Dynam. Syst. 5, 285–289 (1987)
Singer, D.: Stable orbits and bifurcations of maps of the interval. SIAM J. Appl. Math. 35, 260–267 (1978)
Sullivan, D. : Class notes at the CUNY Graduate from 1986–1990
Sullivan, D.: Differentiable structures on fractal-like sets determined by intrinsic scaling functions on dual Cantor sets. The Mathematical Heritage of Hermann Weyl. A.M.S. Proc. Symp. Pure Math. 48, 15–23 (1987)
Tukia, P.: Differentiability and rigidity of Möbius groups. Invent. Math. 82, 557–578 (1985)
Acknowledgements
This survey article is written under an invitation by Professor Alberto Pinto in the occasion of the 60th birthday of Professor David Rand. The author would like to use this opportunity to thank Professor Rand for kindness help when he just graduated from the CUNY Graduate Center. Professors Rand and Pinto have been worked in this direction and made a contribution to this direction. The reader who is interested in this direction and their work can go to their survey articles in this volume (see also [1]). The author would like to use this opportunity to thank Professor Dennis Sullivan and Professor Fred Gardiner for their lectures and for their support and help for many years during this research. The author’s work in this direction has been partially supported by grants from NSF and awards from PSC-CUNY and grants from AMSS and MCM at Chinese Academy of Sciences.
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Jiang, Y. (2011). Differential Rigidity and Applications in One-Dimensional Dynamics. In: Peixoto, M., Pinto, A., Rand, D. (eds) Dynamics, Games and Science I. Springer Proceedings in Mathematics, vol 1. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11456-4_31
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DOI: https://doi.org/10.1007/978-3-642-11456-4_31
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