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

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

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

Maja Resman ^1,

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

Received July 2012 Revised September 2012 Published January 2013

Abstract
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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.

Keywords: Minkowski content., formal classification, Parabolic diffeomorphisms, box dimension.

Mathematics Subject Classification: 28A75, 28A78, 37C05, 37C10, 37C15, 37C2.

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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