By Alessandrini L.

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S. Pontryagin and L. G. Shnirelman in 1932. It can be shown that any compact set with the finite fractal dimension is homeomorphic to a subset of the space R d when d > 0 is large enough. To obtain the estimates of the fractal dimension the following simple assertion is useful. 1. The following equality holds: ln N ( M , e ) dimf M = lim , e ® 0 ln ( 1 ¤ e ) where N ( M , e ) is the cardinality of the minimal covering of the compact M with closed sets diameter of which does not exceed 2 e (the diameter of a set X is defined by the value d ( X ) = sup { x - y : x , y Î X } ).

E. which lie at a positive distance from the line { v = 0 , w = 0 } . e. sup { dist ( St y , Cn ) , y Î B } £ C e for t ³ tB , where g is a positive constant. -g ( t - tB ) 43 44 Basic Concepts of the Theory of Infinite-Dimensional Dynamical Systems C h a p t e r Now let y0 = ( u0 , v0 , w0 ) lie in the global attractor A of the system ( R 3 , St ) . Assume that r 0 ¹ 0 and r 02 = v02 + w02 ¹ - m n . 1) there exists a trajectory g = { y ( t ) = ( u ( t ) ; v ( t ) ; w ( t ) ) , t Î R } lying in A such that y (0) = y0 .

Thus, we consider the covering { B ( xi , 2 e ( q + d + g ) ) , i = 1, 2, ¼, N (e, q, d) } of the set VAq . It is evident that every point x Î VAq belongs to this covering together with the ball B ( x , 2 g e ) . If j ³ 2 , the inequalities Finite Dimensionality of Invariant Sets h ( Ej , VAq ) £ h ( V j M , VA q ) £ h ( V jM , V Ej -1 ) hold. 2) we have that h ( V j M , VEj - 1 ) £ ( l + d ) h ( V j - 1 M , Ej - 1 ) £ ( l + d ) q j - 1 . e. if l + d ln 2ge j ³ j0 º 2 + .

### 1-Convex Manifolds are p-Kahler by Alessandrini L.

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