On cocycles with values in the group SU(2)

In this paper we introduce the notion of degree for $C^1$-cocycles over irrational rotations on the circle with values in the group SU(2). It is shown that if a $C^1$-cocycle $\phi:S^1\to SU(2)$ over an irrational rotation by $\alpha$ has nonzero degree, then the skew product $S^1\times SU(2)\ni(x,g)\mapsto (x+\alpha,g\phi(x))\in S^1\times SU(2)$ is not ergodic and the group of essential values of $\phi$ is equal to the maximal Abelian subgroup of SU(2). Moreover, if $\phi$ is of class $C^2$ (with some additional assumptions) the Lebesgue component in the spectrum of the skew product has countable multiplicity. Possible values of degree are discussed, too.Comment: 30 page

Ergodic properties of infinite extensions of area-preserving flows

We consider volume-preserving flows $(\Phi^f_t)_{t\in\mathbb{R}}$ on $S\times \mathbb{R}$, where $S$ is a closed connected surface of genus $g\geq 2$ and $(\Phi^f_t)_{t\in\mathbb{R}}$ has the form $\Phi^f_t(x,y)=(\phi_tx,y+\int_0^t f(\phi_sx)ds)$, where $(\phi_t)_{t\in\mathbb{R}}$ is a locally Hamiltonian flow of hyperbolic periodic type on $S$ and $f$ is a smooth real valued function on $S$. We investigate ergodic properties of these infinite measure-preserving flows and prove that if $f$ belongs to a space of finite codimension in $\mathscr{C}^{2+\epsilon}(S)$, then the following dynamical dichotomy holds: if there is a fixed point of $(\phi_t)_{t\in\mathbb{R}}$ on which $f$ does not vanish, then $(\Phi^f_t)_{t\in\mathbb{R}}$ is ergodic, otherwise, if $f$ vanishes on all fixed points, it is reducible, i.e. isomorphic to the trivial extension $(\Phi^0_t)_{t\in\mathbb{R}}$. The proof of this result exploits the reduction of $(\Phi^f_t)_{t\in\mathbb{R}}$ to a skew product automorphism over an interval exchange transformation of periodic type. If there is a fixed point of $(\phi_t)_{t\in\mathbb{R}}$ on which $f$ does not vanish, the reduction yields cocycles with symmetric logarithmic singularities, for which we prove ergodicity.Comment: 57 pages, 4 picture

Cocycles over interval exchange transformations and multivalued Hamiltonian flows

We consider interval exchange transformations of periodic type and construct different classes of recurrent ergodic cocycles of dimension $\geq 1$ over this special class of IETs. Then using Poincar\'e sections we apply this construction to obtain recurrence and ergodicity for some smooth flows on non-compact manifolds which are extensions of multivalued Hamiltonian flows on compact surfaces.Comment: 45 pages, 2 figure

Non-reversibility and self-joinings of higher orders for ergodic flows

By studying the weak closure of multidimensional off-diagonal self-joinings we provide a criterion for non-isomorphism of a flow with its inverse, hence the non-reversibility of a flow. This is applied to special flows over rigid automorphisms. In particular, we apply the criterion to special flows over irrational rotations, providing a large class of non-reversible flows, including some analytic reparametrizations of linear flows on the two torus, so called von Neumann's flows and some special flows with piecewise polynomial roof functions.. A topological counterpart is also developed with the full solution of the problem of the topological self-similarity of continuous special flows over irrational rotations. This yields examples of continuous special flows over irrational rotations without topological self-similarities and having all non-zero real numbers as scales of measure-theoretic self-similarities.Comment: 49 pages, 2 figur