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 and Continuous Dynamical Systems, 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., Amsterdam, 1958.

[2]

J. Ecalle, "Les Fonctions Résurgentes. Tome III," Publications Mathématiques d'Orsay, 85, Université de Paris-Sud, Département de Mathematiques, Orsay, 1985.

[3]

K. Falconer, "Fractal Geometry: Mathematical Foundations and Applications," John Wiley and Sons Ltd., Chichester, 1990.

[4]

Y. Ilyashenko and S. Yakovenko, "Lectures on Analytic Differential Equations, Graduate Studies in Mathematics," 86 American Mathematical Society, Providence, RI, 2008, xiv+625.

[5]

I. Kluvanek and G. Knowles, "Vector Measures and Control Systems," North-Holland Mathematics Studies 20, Amsterdam, 1976.

[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, AL, 1990), Math. Sci. Engrg., 186 (1992), Academic Press, Boston, 151-181. doi: 10.1016/S0076-5392(08)63379-2.

[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), 41-69. doi: 10.1112/plms/s3-66.1.41.

[8]

F. Loray, "Pseudo-Groupe D'une Singularité de Feuilletage Holomorphe en Dimension Deux," Prépublication IRMAR, ccsd-00016434, 2005.

[9]

P. Mardešić, M. Resman and V. Županović, Multiplicity of fixed points and growth of $\varepsilon$-neighborhoods of orbits, J. Differential Equations, 253 (2012), 2493-2514. doi: 10.1016/j.jde.2012.06.020.

[10]

J. Milnor, "Dynamics in One Complex Variable, Introductory Lectures," $2^{nd}$ edition, Friedr. Vieweg. & Sohn Verlagsgesellschaft mbH, Braunschweig/Wiesbaden, 1999.

[11]

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

[12]

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

[13]

C. Tricot, "Curves and Fractal Dimension," Springer-Verlag, Paris, 1993.

[14]

V. Županović and D. Žubrinić, Fractal dimensions in dynamics, in "Encyclopedia of Mathematical Physics" 2 (2006), Elsevier, Oxford, 394-402.

[15]

S. M. Voronin, Analytic classification of germs of conformal mappings $(\mathbbC,0)\to(\mathbbC,0)$, Functional Anal. Appl., 15(1981), 1-13.

show all references

References:
[1]

N. G. de Bruijn, "Asymptotic Methods in Analysis," North-Holland Publishing Co., Amsterdam, 1958.

[2]

J. Ecalle, "Les Fonctions Résurgentes. Tome III," Publications Mathématiques d'Orsay, 85, Université de Paris-Sud, Département de Mathematiques, Orsay, 1985.

[3]

K. Falconer, "Fractal Geometry: Mathematical Foundations and Applications," John Wiley and Sons Ltd., Chichester, 1990.

[4]

Y. Ilyashenko and S. Yakovenko, "Lectures on Analytic Differential Equations, Graduate Studies in Mathematics," 86 American Mathematical Society, Providence, RI, 2008, xiv+625.

[5]

I. Kluvanek and G. Knowles, "Vector Measures and Control Systems," North-Holland Mathematics Studies 20, Amsterdam, 1976.

[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, AL, 1990), Math. Sci. Engrg., 186 (1992), Academic Press, Boston, 151-181. doi: 10.1016/S0076-5392(08)63379-2.

[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), 41-69. doi: 10.1112/plms/s3-66.1.41.

[8]

F. Loray, "Pseudo-Groupe D'une Singularité de Feuilletage Holomorphe en Dimension Deux," Prépublication IRMAR, ccsd-00016434, 2005.

[9]

P. Mardešić, M. Resman and V. Županović, Multiplicity of fixed points and growth of $\varepsilon$-neighborhoods of orbits, J. Differential Equations, 253 (2012), 2493-2514. doi: 10.1016/j.jde.2012.06.020.

[10]

J. Milnor, "Dynamics in One Complex Variable, Introductory Lectures," $2^{nd}$ edition, Friedr. Vieweg. & Sohn Verlagsgesellschaft mbH, Braunschweig/Wiesbaden, 1999.

[11]

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

[12]

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

[13]

C. Tricot, "Curves and Fractal Dimension," Springer-Verlag, Paris, 1993.

[14]

V. Županović and D. Žubrinić, Fractal dimensions in dynamics, in "Encyclopedia of Mathematical Physics" 2 (2006), Elsevier, Oxford, 394-402.

[15]

S. M. Voronin, Analytic classification of germs of conformal mappings $(\mathbbC,0)\to(\mathbbC,0)$, Functional Anal. Appl., 15(1981), 1-13.

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