On estimations of generalized Hausdorff dimension
DOI:
https://doi.org/10.21638/11701/spbu01.2019.401Abstract
The definition of an abstract homogeneous dimensional space with finite index of compactness is given, as well as the definition of Hausdorff — Besicovitch dimensional spectrum of such a space. The possible values of the last one are studied. Also some abstract version of Duady — Oesterle theorem is given.
Keywords:
Hausdorff — Lebesgue measure-like functional, homogeneous dimensional space with finite index of compactnes, Hausdorff — Besicovitch dimensional spectrum
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References
Литература
Hausdorff F. Dimension und ¨außere Maß // Mathematische Annalen. 1919. Vol. 79. P. 157–179. https://doi.org/10.1007/BF01457179
Boichenko V.A., Leonov G.A., Reitmann V. Dimension Theory for Ordinary Differential Equations. Wiesbaden: Teubner, 2005. https://doi.org/10.1007/978-3-322-80055-8
Leonov G.A. On Estimation of the Hausdorff Dimension of Attractors // Vestnik St. Petersburg University, Mathematics. 1991. Vol. 24, No. 3. P. 41.
Blincherskaya M.A., Ilyashenko Y.S. Estimate for the entropy dimension if the maximal attractor for k-contracting systems in an infinity dimensional space // Russian Journal of Math. Phys. 1999. Vol. 6, No. 1. P. 20–26.
Barreira L., Gelfert K. Dimension estimates in smooth dynamics: a survey of recent results // Ergodic Theory and Dynamical Systems. 2011. Vol. 31, No. 03. P. 641–671. https://doi.org/10.1017/S014338571000012X
Kuznetsov N.V. The Lyapunov dimension and its estimation via the Leonov method // Physics Letters A. 2016. Vol. 380, No. 25–26. P. 2142–2149. https://doi.org/10.1016/j.physleta.2016.04.036
Leonov G.A., Kuznetsov N.V., Korzhemanova N.A., Kusakin D.V. Lyapunov dimension formula for the global attractor of the Lorenz system // Communications in Nonlinear Science and Numerical Simulation. 2016. Vol. 41. P. 84–103. https://doi.org/10.1016/j.cnsns.2016.04.032
Kuznetsov N.V., Alexeeva T.A., Leonov G.A. Invariance of Lyapunov exponents and Lyapunov dimension for regular and irregular linearizations // Nonlinear Dynamics. 2016. Vol. 85, No. 1. P. 195–201. https://doi.org/10.1007/s11071-016-2678-4
Leonov G.A. Lyapunov functions in the attractors dimension theory // Appl. Math. and Mech. 2012. Vol. 76. P. 129–141. https://doi.org/10.1016/j.jappmathmech.2012.05.002
Leonov G.A. Strange Attractors and Classical Stability Theory. St. Petersburg: St. Petersburg Univ. Press, 2008.
Leonov G.A. Formulas for the Lyapunov dimension of attractors of the generalized Lorenz system // Doklady mathematics. 2013. Vol. 450, No. 1. P. 13–18.
Leonov G.A. Hausdorff — Lebesgue Dimension of Attractors // International Journal of Bifurcations and Chaos. 2017. Vol. 27, No. 10. Art. no. 1750164. https://doi.org/10.1142/S0218127417501644
Besicovitch A.S. On existence of subsets of finite measure of sets of infinite measure // Indagat. muth. 1952. Vol. 14. P. 339–344. https://doi.org/10.1016/S1385-7258(52)50045-3
Rogers C.A. Hausdorff measures. Cambridge University Press, 1998.
Humke P.D., Petruska G. The packing dimension of a typical continuous function is 2 // Real Analysis Exchange. 1988. Vol. 14. P. 345–357. https://doi.org/10.2307/44151950
Hunt B. Maximum local Lyapunov dimension bounds the box dimensions of a chaotic attractors // Nonlinearity. 1996. Vol. 9. P. 845–852. https://doi.org/10.1088/0951-7715/9/4/001
Douady A., Oesterle I. Dimension de Hausdorff des Attractors // C.R. Acad. Sei. Paris. Ser. A. 1980. Vol. 290, No. 24. P. 1135–1138.
References
Hausdorff F., “Dimension und ¨außere Maß”, Mathematische Annalen 79, 157–179 (1919). https://doi.org/10.1007/BF01457179
Boichenko V.A., Leonov G.A., Reitmann V., Dimension Theory for Ordinary Differential Equations, (Teubner, Wiesbaden, 2005). https://doi.org/10.1007/978-3-322-80055-8
Leonov G.A., “On Estimation of the Hausdorff Dimension of Attractors”, Vestnik St. Petersburg University, Mathematics 24(3), 41 (1991).
Blincherskaya M.A., Ilyashenko Y. S., “Estimate for the entropy dimension if the maximal attractor for k-contracting systems in an infinity dimensional space”, Russian Journal of Math. Phys. 6(1), 20–26 (1999).
Barreira L., Gelfert K., “Dimension estimates in smooth dynamics: a survey of recent results”, Ergodic Theory and Dynamical Systems 31(03), 641–671 (2011). https://doi.org/10.1017/S014338571000012X
Kuznetsov N.V., “The Lyapunov dimension and its estimation via the Leonov method”, Physics Letters A 380(25–26), 2142–2149 (2016). https://doi.org/10.1016/j.physleta.2016.04.036
Leonov G.A., Kuznetsov N.V., Korzhemanova N.A., Kusakin D.V., “Lyapunov dimension formula for the global attractor of the Lorenz system”, Communications in Nonlinear Science and Numerical Simulation 41, 84–103 (2016). https://doi.org/10.1016/j.cnsns.2016.04.032
Kuznetsov N.V., Alexeeva T.A., Leonov G.A., “Invariance of Lyapunov exponents and Lyapunov dimension for regular and irregular linearizations”, Nonlinear Dynamics 85(1), 195–201 (2016). https://doi.org/10.1007/s11071-016-2678-4
Leonov G.A., “Lyapunov functions in the attractors dimension theory”, Appl. Math. and Mech. 76, 129–141 (2012). https://doi.org/10.1016/j.jappmathmech.2012.05.002
Leonov G.A., Strange Attractors and Classical Stability Theory (St. Petersburg Univ. Press, St. Petersburg, 2008).
Leonov G.A., “Formulas for the Lyapunov dimension of attractors of the generalized Lorenz system”, Doklady mathematics 450(1), 13–18 (2013).
Leonov G.A., “Hausdorff — Lebesgue Dimension of Attractors”, International Journal of Bifurcations and Chaos 27(10), 1750164 (2017). https://doi.org/10.1142/S0218127417501644
Besicovitch A. S., “On existence of subsets of finite measure of sets of infinite measure”, Indagat. muth. 14, 339–344 (1952). https://doi.org/10.1016/S1385-7258(52)50045-3
Rogers C.A., Hausdorff measures (Cambridge University Press, 1998).
Humke P.D., Petruska G., “The packing dimension of a typical continuous function is 2”, Real Analysis Exchange 14, 345–357 (1988). https://doi.org/10.2307/44151950
Hunt B., “Maximum local Lyapunov dimension bounds the box dimensions of a chaotic attractors”, Nonlinearity 9, 845–852 (1996). https://doi.org/10.1088/0951-7715/9/4/001
Douady A., Oesterle I., “Dimension de Hausdorff des Attractors”, C.R. Acad. Sei. Paris. Ser. A 290(24), 1135–1138 (1980).
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Published
2019-11-28
How to Cite
Leonov, G. A., & Florynskii, A. A. (2019). On estimations of generalized Hausdorff dimension. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 6(4), 534–543. https://doi.org/10.21638/11701/spbu01.2019.401
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In memoriam of G. A. Leonov
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Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.
