Abstract
We study the behaviour of Minkowski content of bounded sets under bi-Lipschitzian mappings. Applications include Minkowski contents and box dimensions of spirals in ℝ3, dynamical systems, and singular integrals.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M. Caubergh, F. Dumortier,Hopf-Takens bifurcations and centers, J. Differential Equations,202 (2004), no. 1, 1–31.
M. Caubergh, J.P. Françoise,Generalized Liénard equations, cyclicity and Hopf-Takens bifurcations, Qualitative Theory of Dynamical Systems6 (2005), 195–222.
Y. Dupain, M. Mendès France, C. Tricot,Dimension de spirales, Bull. Soc. Math. France111 (1983), 193–201.
L.C. Evans, R.F. Gariepy,Measure theory and fine properties of functions, CRC Press, 1992.
K. Falconer,Fractal Geometry, Chichester: Wiley (1990).
A. Gray,Tubes, Addison Wesley, 1990.
J. Guckenheimer, P. Holmes,Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer Verlag, 1983.
C.Q. He, M.L. Lapidus,Generalized Minkowski content, spectrum of fractal drums, fractal strings and the Riemann zeta-function. Mem. Amer. Math. Soc.127 (1997), no. 608.
L. Horvat, D. Žubrinić,Maximally singular Sobolev functions, J. Math. Anal. Appl.,304 (2005), no. 2, 531–541.
M. Lapidus, M. van Frankenhuysen,Fractal Geometry and Number Theory. Complex dimensions of fractal strings and zeros of zeta functions. Birkhäuser Boston, Inc., Boston, MA, 2000.
P. Mattila,Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability, Cambridge 1995.
M. Pašić,Minkowski-Bouligand dimension of solutions of the one-dimensional p-Laplacian, J. Differential Equations190 (2003) 268–305.
M. Pašić, V. Županović,Some metric-singular properties of the graph of solutions of the one-dimensional p-Laplacian, Electronic J. of Differential Equations60 (2004), 1–25.
F. Takens,Unfoldings of certain singularities of vector fields: Generalized Hopf bifurcations, J. Differential Equations,14 (1973), 476–493.
C. Tricot,Curves and Fractal Dimension, Springer-Verlag, 1995.
D. Žubrinić,Singular sets of Sobolev functions, C. R. Acad. Sci., Paris, Série 1,334 (2002), 539–544.
D. Žubrinić,Singular sets of Lebesgue integrable functions, Chaos, Solitons, Fractals,21 (2004), 1281–1287.
D. Žubrinić,Analysis of Minkowski contents of fractal sets and applications, Real Anal. Exchange, Vol31 (2), 2005/2006, 315–354.
D. Žubrinić, V. Županović,Fractal analysis of spiral trajectories of some planar vector fields, Bulletin des Sciences Mathématiques,129/6 (2005), 457–485.
D. Žubrinić, V. Županović,Fractal analysis of spiral trajectories of some vector fields in ℝ3, C. R. Acad. Sci. Paris, Série I, Vol.342, 12 (2006), 959–963.
D. Žubrinić, D. Žubrinić,Fractal dimensions in dynamics, inEncyclopedia of Mathematical Physics, eds. J.-P. Françoise, G.L. Naber and Tsou S.T. Oxford: Elsevier, 2006, vol 2, 394–402.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Žubrinić, D., Županović, V. Box dimension of spiral trajectories of some vector fields in ℝ3 . Qual. Th. Dyn. Syst. 6, 251–272 (2005). https://doi.org/10.1007/BF02972676
Issue Date:
DOI: https://doi.org/10.1007/BF02972676