1. We answer Michael Gordins question providing singular spectrum for transformations with homoclinic Bernoulli flows via Poisson suspensions induced by modified Sidon rank-one constructions. 2. We give homoclinic proof of Emmanuel Roys theorem on multiple mixing of Poisson suspensions, adding new examples to Jonathan Kings ergodic homoclinic groups of special zero-entropy transformations. 3. Sasha Prikhodko found the fast decay of correlations for some iceberg automorphisms. We get similar correlations for a class of infinite rank-one Sidon transformations. This version is based on On Mixing Rank One Infinite Transformations arXiv:1106.4655
In real Hilbert spaces, this paper generalizes the orthogonal groups $mathrm{O}(n)$ in two ways. One way is by finite multiplications of a family of operators from reflections which results in a group denoted as $Theta(kappa)$, the other is by considering the automorphism group of the Hilbert space denoted as $O(kappa)$. We also try to research the algebraic relationship between the two generalizations and their relationship to the stable~orthogonal~group~$mathrm{O}=varinjlimmathrm{O}(n)$ in terms of topology. In this paper we mainly show that : (a) $Theta(kappa)$ is a topological and normal subgroup of $O(kappa)$; (b) $O^{(n)}(kappa) to O^{(n+1)}(kappa) stackrel{pi}{to} S^{kappa}$ is a fibre bundle where $O^{(n)}(kappa)$ is a subgroup of $O(kappa)$ and $S^{kappa}$ is a generalized sphere.
Global resonance is a mechanism by which a homoclinic tangency of a smooth map can have infinitely many asymptotically stable, single-round periodic solutions. To understand the bifurcation structure one would expect to see near such a tangency, in this paper we study one-parameter perturbations of typical globally resonant homoclinic tangencies. We assume the tangencies are formed by the stable and unstable manifolds of saddle fixed points of two-dimensional maps. We show the perturbations display two infinite sequences of bifurcations, one saddle-node the other period-doubling, between which single-round periodic solutions are asymptotically stable. Generically these scale like $|lambda|^{2 k}$, as $k to infty$, where $-1 < lambda < 1$ is the stable eigenvalue associated with the fixed point. If the perturbation is taken tangent to the surface of codimension-one homoclinic tangencies, they instead scale like $frac{|lambda|^k}{k}$. We also show slower scaling laws are possible if the perturbation admits further degeneracies.
We prove that every $C^1$ three-dimensional flow with positive topological entropy can be $C^1$ approximated by flows with homoclinic orbits. This extends a previous result for $C^1$ surface diffeomorphisms cite{g}.
We prove that any diffeomorphism of a compact manifold can be C^1-approximated by a diffeomorphism which exhibits a homoclinic bifurcation (a homoclinic tangency or a heterodimensional cycle) or by a diffeomorphism which is partially hyperbolic (its chain-recurrent set splits into partially hyperbolic pieces whose centre bundles have dimensions less or equal to two). We also study in a more systematic way the central models introduced in arXiv:math/0605387.