By Igor Nikolaev (auth.)

Foliations is among the significant techniques of contemporary geometry and topology which means a partition of topological house right into a disjoint sum of leaves. This e-book is dedicated to geometry and topology of floor foliations and their hyperlinks to ergodic idea, dynamical platforms, complicated research, differential and noncommutative geometry. This finished publication addresses graduate scholars and researchers and should function a reference publication for specialists within the field.

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Additional info for Foliations on Surfaces

Example text

We see at once how 42 2. Morse-Smale Foliations to extend the homeomorphism h to M*\8M* via the equation hetz = (thz, where z E 8V U {qi}. 1 One easily sees that h is invertible (the mapping h- 1 can be constructed in the same manner, replacing by (). It remains to prove that h is continuous. Our proof is much the same as in [234J. The continuity of h is evident at sinks, sources and in the neighbourhoods of saddles and saddle elements. We shall prove the continuity of h at the points of stable separatrices of the saddle elements.

A 'singularity scheme' is the equivalence class of words for a relation E. The following statement, proved in [92], is a sufficient condition of CO-equivalence. 2 (Takens-Dumortier) If(X,L i , Wd and (Y,L 2 , W 2 ) are two triples defined as above, and if they have the same 'singularity scheme', Wi rv W 2 , different from the word C;:, then there exists a homeomorphism h from the neighbourhood U of the singularity X(O) of the vector field X to the neighbourhood V of the singularity Y(O) of the vector field Y which conjugates the orbits of X and Y near 0, preserving their orientation.

19) takes the form ~~ = -X1/ + D2x 3, ~~ dB 2 dt = -B + B = -1/ + 1/ 2 - D2X21/ + D2B4 y 2 - 2 C 2By , dy dt + C2X21/ 4; 3 = -By + C 2y . 20). Ml is an elementary point (not hyperbolic) and is a topological node when D2 < 0, and a topological saddle when D2 > O. The point M2 is a hyperbolic saddle. 21) is an elementary point with one eigenvalue equal to zero. This is a node if C2 < 0 and a saddle if C2 > O. 19) is unfolded, and it is a weak (ii)-singularity when D2 > 0, C 2 > 0; a weak (i)-singularity when D 2C 2 < 0; and a weak (iii)-singularity when D2 < 0, C 2 < O.