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Proof. It suffices to prove the assertions for g D 1 2 G. Indeed, for every g 2 G the map g W H ! HI /, and hence maps recurrent points to recurrent points. x0 / : Since the projection W H ! x0 /. x0 /. x0 ; 1/. x0 ; 1/ for all n 2 N. GI h/, so if V is a neighborhood of 1, then hn 2 V for some n 2 N. x0 ; hn / 2 U V for any neighborhood U of x0 . HI /. An analogous result is true for uniformly recurrent points. 16. HI / the group extension along ˚ W K ! G. x0 ; g/ 2 H is uniformly recurrent in H for all g 2 G.

1; 1/. 36 the product system is not topologically transitive (cf. 2). Fig. 38 and its orbit closure, and the same for the cases a1 D e5i , a2 D e2i and a1 D e5i , a2 D e8i 26 2 Topological Dynamical Systems Because of these two examples, it is interesting to characterize transitivity of the rotations on the d-torus for d > 1. This is a classical result of Kronecker (1885). 39 (Kronecker). a1 ; : : : ; ad / 2 Td . Td I a/ is topologically transitive if and only if a1 ; a2 ; : : : ; ad are linearly independent in the Z-module T (which means that if ak11 ak22 akdd D 1 for k1 ; k2 ; : : : ; kd 2 Z, then k1 D k2 D D kd D 0).

K= I '/ ! LI /; is an isomorphism of the two systems. 18 (Homogeneous Systems II). GI a/ and let be a closed subgroup of G. The equivalence relation x Def. ax/ D y 1 a 1 ax D y 1 x. The set of corresponding equivalence classes is simply the homogeneous space ˚ D g G= W g2G « of left cosets, and the induced dynamics on it is given by g 7! ag . G= I a/, cf. 11. GI a/ ! g/ WD g is a factor map of topological dynamical systems. 19 (Group Factors). KI '/. Consider the equivalence relation x H y Def.

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Extremal functions for Moser-Trudinger type inequality on compact closed 4-manifolds by Li Y.Y., Ndiaye C.B.

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