Link to this article: http://journals.cambridge.org/abstract_S0143385706000836How to cite this article: CORINNA ULCIGRAI (2007). Mixing of asymmetric logarithmic suspension ows over interval exchange transformations.Abstract. We consider suspension flows built over interval exchange transformations with the help of roof functions having an asymmetric logarithmic singularity. We prove that such flows are strongly mixing for a full measure set of interval exchange transformations.

We consider a symbolic coding of linear trajectories in the regular octagon with opposite sides identified (and more generally in regular 2n-gons). Each infinite trajectory gives a cutting sequence corresponding to the sequence of sides hit. We give an explicit characterization of these cutting sequences. The cutting sequences for the square are the well-studied Sturmian sequences which can be analysed in terms of the continued fraction expansion of the slope. We introduce an analogous continued fraction algorithm which we use to connect the cutting sequence of a trajectory with its slope. Our continued fraction expansion of the slope gives an explicit sequence of substitution operations which generate the cutting sequences of trajectories with that slope. Our algorithm can be understood in terms of renormalization of the octagon translation surface by elements of the Veech group.Moreover, the converse of Proposition 1.2.2 is almost true; the exceptions, that is, words in {A, B} Z which are infinitely derivable and are not cutting sequences such as . . . BBBBBABBBBB . . ., can be explicitly described. The space of words has a natural topology which makes it a compact space. The word given above is not a cutting sequence, but it has the property that any finite subword can be realized by a finite trajectory. This is equivalent to saying that it is in the closure of the space of cutting sequences. The closure of the space of cutting sequences is precisely the set of infinitely derivable sequences.Proof of Proposition 1.2.2. Let us prove that the derived sequence w of a cutting sequence w = c(τ ) is again a cutting sequence of a linear trajectory on the square. Since a cutting sequence is admissible, this will suffice to show that every derived sequence is again admissible. † In this section, we are using the terminology from [18].

We consider smooth time-changes of the classical horocycle flows on the unit tangent bundle of a compact hyperbolic surface and prove sharp bounds on the rate of equidistribution and the rate of mixing. We then derive results on the spectrum of smooth time-changes and show that the spectrum is absolutely continuous with respect to the Lebesgue measure on the real line and that the maximal spectral type is equivalent to Lebesgue.

We give a criterion which allows to prove non-ergodicity for certain infinite periodic billiards and directional flows on Z-periodic translation surfaces. Our criterion applies in particular to a billiard in an infinite band with periodically spaced vertical barriers and to the Ehrenfest wind-tree model, which is a planar billiard with a Z 2 -periodic array of rectangular obstacles. We prove that, in these two examples, both for a full measure set of parameters of the billiard tables and for tables with rational parameters, for almost every direction the corresponding billiard flow is not ergodic and has uncountably many ergodic components. As another application, we show that for any recurrent Z-cover of a square tiled surface of genus two the directional flow is not ergodic and has no invariant sets of finite measure for a full measure set of directions. In the language of essential values, we prove that the skewproducts which arise as Poincaré maps of the above systems are associated to non-regular Z-valued cocycles for interval exchange transformations.

In this paper we prove results on Birkhoff and Oseledets genericity along certain curves in the space of affine lattices and in moduli spaces of translation surfaces. We also prove applications of these results to dynamical billiards, mathematical physics and number theory. In the space of affine lattices ASL 2 (R)/ASL 2 (Z), we prove that almost every point on a curve with some non-degeneracy assumptions is Birkhoff generic for the geodesic flow. This implies almost everywhere genericity for some curves in the locus of branched covers of the torus inside the stratum H(1, 1) of translation surfaces. For these curves (and more in general curves which are well-approximated by horocycle arcs and satisfy almost everywhere Birkhoff genericity) we also prove that almost every point is Oseledets generic for the Kontsevitch-Zorich cocycle, generalizing a recent result by Chaika and Eskin. As applications, we first consider a class of pseudo-integrable billiards, billiards in ellipses with barriers, which was recently explored by Dragović and Radnović, and prove that for almost every parameter, the billiard flow is uniquely ergodic within the region of phase space in which it is trapped. We then consider any periodic array of Eaton retroreflector lenses, placed on vertices of a lattice, and prove that in almost every direction light rays are each confined to a band of finite width. This generalizes a phenomenon recently discovered by Frączek and Schmoll which could so far only be proved for random periodic configurations. Finally, a result on the gap distribution of fractional parts of the sequence of square roots of positive integers, which extends previous work by Elkies and McMullen, is also obtained.

We consider volume-preserving flows (Φ f t ) t∈R on S × R, where S is a compact connected surface of genus g ≥ 2 and (Φwhere (φ t ) t∈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 C 2+ (S), then the following dynamical dichotomy holds: if there is a fixed point of (φ t ) t∈R on which f does not vanish, then (Φ f t ) t∈R is ergodic, otherwise, if f vanishes on all fixed points, it is reducible, i.e. isomorphic to the trivial extension (Φ 0 t ) t∈R . The proof of this result exploits the reduction of (Φ f t ) t∈R to a skew product automorphism over an interval exchange transformation of periodic type. If there is a fixed point of (φ t ) t∈R on which f does not vanish, the reduction yields cocycles with symmetric logarithmic singularities, for which we prove ergodicity.

We prove that minimal area-preserving flows locally given by a smooth Hamiltonian on a closed surface of genus g ≥ 2 are typically (in the measure-theoretical sense) not mixing. The result is obtained by considering special flows over interval exchange transformations under roof functions with symmetric logarithmic singularities and proving absence of mixing for a full measure set of interval exchange transformations. Definitions and main results1.1. Flows given by multi-valued Hamiltonians. Let us consider the following natural construction of area-preserving flows on surfaces. On a closed connected orientable surface S of genus g ≥ 1 with a fixed smooth area form ω, consider a smooth closed real-valued differential 1-form η. Let X be the vector field determined by η = i X ω = ω(η, ·) and consider the flow {ϕ t } t∈R on S associated to X. Since η is closed, the transformations ϕ t , t ∈ R, are areapreserving. The flow {ϕ t } t∈R is known as the multi-valued Hamiltonian flow associated to η. Indeed, the flow {ϕ t } t∈R is locally Hamiltonian; i.e., locally one can find coordinates (x, y) on S in which it is given by the solution to the equationsẋ = ∂H/∂y,ẏ = −∂H/∂x for some smooth real-valued Hamiltonian function H. A global Hamiltonian H cannot be in general be defined (see [NZ99, §1.3.4]), but one can think of {ϕ t } t∈R as globally given by a multi-valued Hamiltonian function.The study of flows given by multi-valued Hamiltonians was initiated by S. P. Novikov [Nov82] in connection with problems arising in solid-state physics i.e., the motion of an electron in a metal under the action of a magnetic field. The orbits of such flows arise also in pseudo-periodic topology, as hyperplane sections of periodic surfaces in T n (see e.g. Zorich [Zor99]).From the point of view of topological dynamics, a decomposition into minimal components (i.e., subsurfaces on which the flow is minimal) and periodic components on which all orbits are periodic (elliptic islands around a center and cylinders filled by periodic orbits) was proved independently by Maier

We consider irrational nilflows on any nilmanifold of step at least 2. We show that there exists a dense set of smooth time-changes such that any time-change in this class which is not measurably trivial gives rise to a mixing nilflow. This in particular reproves and generalizes to any nilflow (of step at least 2) the main result proved in [AFU] for the special class of Heisenberg (step 2) nilflows, and later generalized in [Rav2] to a class of nilflows of arbitrary step which are isomorphic to suspensions of higher-dimensional linear toral skew-shifts.