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 - A, 2013, 33 (8) : 3767-3790. doi: 10.3934/dcds.2013.33.3767
References:
[1]

N. G. de Bruijn, "Asymptotic Methods in Analysis,", North-Holland Publishing Co., (1958).   Google Scholar

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J. Ecalle, "Les Fonctions Résurgentes. Tome III,", Publications Mathématiques d'Orsay, 85 (1985).   Google Scholar

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K. Falconer, "Fractal Geometry: Mathematical Foundations and Applications,", John Wiley and Sons Ltd., (1990).   Google Scholar

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Y. Ilyashenko and S. Yakovenko, "Lectures on Analytic Differential Equations, Graduate Studies in Mathematics,", 86 American Mathematical Society, 86 (2008).   Google Scholar

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I. Kluvanek and G. Knowles, "Vector Measures and Control Systems,", North-Holland Mathematics Studies 20, (1976).   Google Scholar

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M. L. Lapidus, Spectral and fractal geometry: From the Weyl-Berry conjecture for the vibrations of fractal drums to the Riemann zeta-function,, Differential Equations and Mathematical Physics (Birmingham, 186 (1992), 151.  doi: 10.1016/S0076-5392(08)63379-2.  Google Scholar

[7]

M. L. Lapidus and C. Pomerance, The Riemann zeta-function and the one-dimensional Weyl-Berry conjecture for fractal drums,, Proceedings of the London Mathematical Society (3), 66 (1993), 41.  doi: 10.1112/plms/s3-66.1.41.  Google Scholar

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F. Loray, "Pseudo-Groupe D'une Singularité de Feuilletage Holomorphe en Dimension Deux,", Prépublication IRMAR, (2005).   Google Scholar

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P. Mardešić, M. Resman and V. Županović, Multiplicity of fixed points and growth of $\varepsilon$-neighborhoods of orbits,, J. Differential Equations, 253 (2012), 2493.  doi: 10.1016/j.jde.2012.06.020.  Google Scholar

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J. Milnor, "Dynamics in One Complex Variable, Introductory Lectures,", $2^{nd}$ edition, (1999).   Google Scholar

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J. Palis and F. Takens, "Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations,", Cambridge University Press, (1993).   Google Scholar

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R. Pratap and A. Ruina, "Introduction to Statistics and Dynamics,", Pre-print for Oxford University Press, (2001).   Google Scholar

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C. Tricot, "Curves and Fractal Dimension,", Springer-Verlag, (1993).   Google Scholar

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V. Županović and D. Žubrinić, Fractal dimensions in dynamics,, in, 2 (2006), 394.   Google Scholar

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S. M. Voronin, Analytic classification of germs of conformal mappings $(\mathbbC,0)\to(\mathbbC,0)$,, Functional Anal. Appl., 15 (1981), 1.   Google Scholar

show all references

References:
[1]

N. G. de Bruijn, "Asymptotic Methods in Analysis,", North-Holland Publishing Co., (1958).   Google Scholar

[2]

J. Ecalle, "Les Fonctions Résurgentes. Tome III,", Publications Mathématiques d'Orsay, 85 (1985).   Google Scholar

[3]

K. Falconer, "Fractal Geometry: Mathematical Foundations and Applications,", John Wiley and Sons Ltd., (1990).   Google Scholar

[4]

Y. Ilyashenko and S. Yakovenko, "Lectures on Analytic Differential Equations, Graduate Studies in Mathematics,", 86 American Mathematical Society, 86 (2008).   Google Scholar

[5]

I. Kluvanek and G. Knowles, "Vector Measures and Control Systems,", North-Holland Mathematics Studies 20, (1976).   Google Scholar

[6]

M. L. Lapidus, Spectral and fractal geometry: From the Weyl-Berry conjecture for the vibrations of fractal drums to the Riemann zeta-function,, Differential Equations and Mathematical Physics (Birmingham, 186 (1992), 151.  doi: 10.1016/S0076-5392(08)63379-2.  Google Scholar

[7]

M. L. Lapidus and C. Pomerance, The Riemann zeta-function and the one-dimensional Weyl-Berry conjecture for fractal drums,, Proceedings of the London Mathematical Society (3), 66 (1993), 41.  doi: 10.1112/plms/s3-66.1.41.  Google Scholar

[8]

F. Loray, "Pseudo-Groupe D'une Singularité de Feuilletage Holomorphe en Dimension Deux,", Prépublication IRMAR, (2005).   Google Scholar

[9]

P. Mardešić, M. Resman and V. Županović, Multiplicity of fixed points and growth of $\varepsilon$-neighborhoods of orbits,, J. Differential Equations, 253 (2012), 2493.  doi: 10.1016/j.jde.2012.06.020.  Google Scholar

[10]

J. Milnor, "Dynamics in One Complex Variable, Introductory Lectures,", $2^{nd}$ edition, (1999).   Google Scholar

[11]

J. Palis and F. Takens, "Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations,", Cambridge University Press, (1993).   Google Scholar

[12]

R. Pratap and A. Ruina, "Introduction to Statistics and Dynamics,", Pre-print for Oxford University Press, (2001).   Google Scholar

[13]

C. Tricot, "Curves and Fractal Dimension,", Springer-Verlag, (1993).   Google Scholar

[14]

V. Županović and D. Žubrinić, Fractal dimensions in dynamics,, in, 2 (2006), 394.   Google Scholar

[15]

S. M. Voronin, Analytic classification of germs of conformal mappings $(\mathbbC,0)\to(\mathbbC,0)$,, Functional Anal. Appl., 15 (1981), 1.   Google Scholar

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