On upper estimates for the Hausdorff dimension of negatively invariant sets of local cocycles. (English. Russian original) Zbl 1303.37009

Vestn. St. Petersbg. Univ., Math. 44, No. 4, 292-300 (2011); translation from Vestn. St-Peterbg. Univ., Ser. I, Mat. Mekh. Astron. 2011, No. 4, 61-70 (2011).
Summary: The paper is concerned with negatively invariant sets of local cocycles generated, in particular, by nonautonomous ordinary differential equations. Upper estimates for the Hausdorff dimension for negatively invariant sets of local cocycles are obtained using singular numbers of linearization of the cocycle and special functions of Lyapunov type.


37C45 Dimension theory of smooth dynamical systems
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
37A20 Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations
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[1] A., Duady and J. Oesterlé, ”Dimension de Hausdorff des Attracteurs,” Comp. Rend. Acad. Sci., Sér. A, No. 290, 1135–1138 (1980).
[2] G. A. Leonov, ”On Estimates of the Hausdorff Dimension of Attractors,” Vestn. Leningr. Gos. Univ. Ser. 1, no. 3, 41–44 (1991). · Zbl 0734.34043
[3] G. A. Leonov and V. A. Boichenko, ”Lyapunov’s Direct Method in the Estimation of the Hausdorff Dimension of Attractors,” Acta Appl. Math. 26, 1–60 (1992). · Zbl 0758.58022
[4] G. A. Leonov, ”Lyapunov Dimension Formulas for Hénon and Lorentz Attractors,” Algebra i Analiz 13(3), 155–170 (2001).
[5] V. A. Boichenko, G. A. Leonov, and V. Reitmann, Dimension Theory for Ordinary Differential Equations (Vieweg-Teubner, Wiesbaden, 2005). · Zbl 1094.34002
[6] G. A. Leonov, Strange Attractors and Classical Stability Theory (St. Petersb. Univ., Saint-Petersburg, 2008) [in Russian].
[7] R. Temam, Infinite-Dimensional Systems in Mechanics and Physics (Springer, New York, 1988). · Zbl 0662.35001
[8] D. R. Wakeman, ”An Application of Topological Dynamics to Obtain a New Invariance Property for Nonautonomous Ordinary Differential Equations,” J. Differ. Equations 17(2), 259–295 (1975). · Zbl 0431.34033
[9] P. E. Kloeden and B. Schmalfuß, ”Nonautonomous Systems, Cocycle Attractors and Variable Time-Step Discretization,” Numer. Algor. 14(1–3), 141–152 (1997). · Zbl 0886.65077
[10] H. Crauel and F. Flandoli, ”Hausdorff Dimension of Invariant Sets for Random Dynamical Systems,” J. Dyn. Differ. Equations 10, 449–474 (1998). · Zbl 0927.37031
[11] M. V. Bebutov, ”On Dynamical Systems in Spaces of Continuous Functions,” Byull. Mekh. Mat. Fak. Mosk. Gos. Univ., No. 5, 1–52 (1941).
[12] O. E. Rössler, ”Different Types of Chaos in Two Simple Differential Equations,” Z. Naturforsch. A. 31, 1664–1670 (1976).
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