August  2013, 33(8): 3767-3790. doi: 10.3934/dcds.2013.33.3767

$\varepsilon$-neighborhoods of orbits and formal classification of parabolic diffeomorphisms

1. 

University of Zagreb, Faculty of Electrical Engineering and Computing, Unska 3, Zagreb, Croatia

Received  July 2012 Revised  September 2012 Published  January 2013

In this article we study the dynamics generated by germs of parabolic diffeomorphisms $f:(\mathbb{C},0)\rightarrow (\mathbb{C},0)$ tangent to the identity. We show how formal classification of a given parabolic diffeomorphism can be deduced from the asymptotic development of what we call directed area of the $\varepsilon$-neighborhood of any orbit near the origin. Relevant coefficients and constants in the development have a geometric meaning. They present fractal properties of the orbit, namely its box dimension, Minkowski content and what we call residual content.
Citation: Maja Resman. $\varepsilon$-neighborhoods of orbits and formal classification of parabolic diffeomorphisms. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3767-3790. doi: 10.3934/dcds.2013.33.3767
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show all references

References:
[1]

North-Holland Publishing Co., Amsterdam, 1958.  Google Scholar

[2]

Publications Mathématiques d'Orsay, 85, Université de Paris-Sud, Département de Mathematiques, Orsay, 1985.  Google Scholar

[3]

John Wiley and Sons Ltd., Chichester, 1990.  Google Scholar

[4]

86 American Mathematical Society, Providence, RI, 2008, xiv+625.  Google Scholar

[5]

North-Holland Mathematics Studies 20, Amsterdam, 1976.  Google Scholar

[6]

Differential Equations and Mathematical Physics (Birmingham, AL, 1990), Math. Sci. Engrg., 186 (1992), Academic Press, Boston, 151-181. doi: 10.1016/S0076-5392(08)63379-2.  Google Scholar

[7]

Proceedings of the London Mathematical Society (3), 66 (1993), 41-69. doi: 10.1112/plms/s3-66.1.41.  Google Scholar

[8]

Prépublication IRMAR, ccsd-00016434, 2005. Google Scholar

[9]

J. Differential Equations, 253 (2012), 2493-2514. doi: 10.1016/j.jde.2012.06.020.  Google Scholar

[10]

$2^{nd}$ edition, Friedr. Vieweg. & Sohn Verlagsgesellschaft mbH, Braunschweig/Wiesbaden, 1999.  Google Scholar

[11]

Cambridge University Press, 1993.  Google Scholar

[12]

Pre-print for Oxford University Press, 2001. Google Scholar

[13]

Springer-Verlag, Paris, 1993.  Google Scholar

[14]

in "Encyclopedia of Mathematical Physics" 2 (2006), Elsevier, Oxford, 394-402. Google Scholar

[15]

Functional Anal. Appl., 15(1981), 1-13.  Google Scholar

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