Global attractors of complete conformal foliations

Transverse Equivalence of Complete Conformal Foliations

“…As was proved in [7,Theorem 5], any complete non-Riemannian conformal foliation (M, F ) of codimension q > 2 is a (Conf (S q ), S q )-foliation and has a global attractor. Let F be the set of foliations covered by fibrations (in the sense of Definition 2.2 below).…”

“…We denote by F C the subset of F consisting of complete conformal non-Riemannian foliations of codimension q > 2 (cf. [7]). …”

“…A conformal foliation (M, F ) is said to be complete if it is a complete Cartan foliation in the sense of [5]. As is shown in [7,Theorem 3], the completeness of a conformal foliation is equivalent to the existence of Ehresmann connections in the sense of Blumenthal-Hebda [8] for this foliation.…”

Dynamics of conformal foliations

“…Since H(L ) is defined up to conjugacy in H, then this definition is correct. We established the following criterion for a conformal foliation to be Riemannian [29,Theorem 3]. [31,Theorem 3], Theorem 3 holds for transversely similar Riemannian foliations of codimension q ≥ 2.…”

“…Note that in [28,29] we use the methods of local and global differential geometry, including foliated principal bundles, geometries on non-Hausdorff manifolds as well as Ehresmann connections for foliations. While B. Deroin and V. Kleptsyn apply methods of random dynamical systems, including Lyapunov exponents of harmonic measures.…”