We consider a symbolic coding of linear trajectories in the regular octagon with opposite sides identified (and more generally 2n-gons). Each infinite trajectory gives cutting sequence corresponding to hit. give an explicit characterization these sequences. The sequences for square are well-studied Sturmian which can be analysed terms continued fraction expansion slope. introduce analogous algorithm we use connect its Our slope substitution operations generate that understood renormalization...

Abstract We consider generalized interval exchange transformations (GIETs) of $$d\ge 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> intervals which are linearizable , i.e. differentiably conjugated to standard maps (IETs) via a diffeomorphism h [0, 1] and study the regularity conjugacy . Using renormalization operator obtained accelerating Rauzy–Veech induction, we show that, under full...

We prove that minimal area-preserving flows locally given by a smooth Hamiltonian on closed surface of genus g ≥ 2 are typically (in the measure-theoretical sense) not mixing.The result is obtained considering special over interval exchange transformations under roof functions with symmetric logarithmic singularities and proving absence mixing for full measure set transformations.

We consider smooth time-changes of the classical horocycle flows onthe unit tangent bundle a compact hyperbolic surface and provesharp bounds on rate equidistribution ofmixing. then derive results spectrum time-changesand show that is absolutely continuous with respect tothe Lebesgue measure real line maximal spectraltype equivalent to Lebesgue.

We consider suspension flows built over interval exchange transformations with the help of roof functions having an asymmetric logarithmic singularity. prove that such are strongly mixing for a full measure set transformations.

We consider reparametrizations of Heisenberg nilflows.We show that if a nilflow is uniquely ergodic, all non-trivial timechanges within dense subspace smooth time-changes are mixing. Equivalently, in the language special flows, we flows over linear skew-shift extensions irrational rotations circle. Without assuming any Diophantine condition on frequency, define class roof functions for which corresponding mixing whenever function not coboundary. Mixing produced by mechanism known as...

Abstract In this paper we prove a renewal-type limit theorem. Given $\alpha \in (0,1)\backslash \mathbb {Q}$ and R &gt;0, let q n be the first denominator of convergents α which exceeds . The main result in is that ratio / has limiting distribution as tends to infinity. existence uses mixing special flow over natural extension Gauss map.

We consider a class of special flows over interval exchange transformations which includes roof functions with symmetric logarithmic singularities. prove that such are typically weakly mixing. As corollary, minimal given by multivalued Hamiltonians on higher-genus surfaces

We give a criterion which allows to prove non-ergodicity for certain infinite periodic billiards and directional flows on Z-periodic translation surfaces. Our applies in particular billiard an band with periodically spaced vertical barriers the Ehrenfest wind-tree model, is planar $Z^2$-periodic array of rectangular obstacles. that, these two examples, both full measure set parameters tables rational parameters, almost every direction corresponding flow not ergodic has uncountably many...

We consider typical area-preserving flows on higher genus surfaces and prove that the flow restricted to mixing minimal components is of all orders, thus answering affirmatively Rokhlin’s multiple question in this context. The main tool a variation Ratner property originally proved by for horocycle flow, i.e. switchable introduced Fayad Kanigowski special over rotations. This property, which independent interest, provides quantitative description parabolic behavior these has implications...

Abstract We prove a rigidity result for foliations on surfaces of genus two, which can be seen as generalization to higher Herman’s theorem circle diffeomorphisms and, correspondingly, flows the torus. in particular that, if smooth, orientable foliation with non-degenerate (Morse) singularities closed surface two is minimal, then, under full measure condition rotation number, it differentiably conjugate linear foliation. The corresponding at level Poincaré sections set (standard) interval...

On considère des marches aléatoires sur la droite réelle, engendrés par rotations irrationnelles, ou, de manière équivalente, produits croisés d'une rotation un nombre réel $\alpha$, dont le cocycle est une fonction constante morceaux moyenne nulle admettant saut à singularité $\beta$. Si $\alpha$ mal approché rationnels et $\beta$ n'est pas bien l'orbite nous démontrons version temporelle du Théorème Limite Centrale (ou Temporal Central Limit theorem dans terminologie qui a été introduite...

We study the size of set ergodic directions for directional billiard flows on infinite band $\R\times [0,h]$ with periodically placed linear barriers length $0

We consider smooth area-preserving flows (also known as locally Hamiltonian ) on surfaces of genus g\geq 1 and study ergodic integrals observables along the flow trajectories. show that these display a power deviation spectrum describe cocycles lead pure behaviour, giving new proof results by Forni [Ann. Math. (2) 155 (2002), 1–103] Bufetov 179 (2014), 431–499] generalizing them to which are non-zero at fixed points. This in particular completes original formulation Kontsevitch–Zorich...

We consider volume-preserving flows $${(\Phi^f_t)_{t\in\mathbb{R}}}$$ on $${S \times \mathbb{R}}$$ , where S is a compact connected surface of genus g ≥ 2 and has the form $${\Phi^f_t(x, y) = (\phi_{t}x, y + \int_0^{t}f(\phi_{s}x)\,ds)}$$ $${(\phi_t)_{t\in\mathbb{R}}}$$ locally Hamiltonian flow hyperbolic periodic type f smooth real valued function S. investigate ergodic properties these infinite measure-preserving prove that if belongs to space finite codimension in...

In this paper we prove results on Birkhoff and Oseledets genericity along certain curves in the space of affine lattices moduli spaces translation surfaces. $ASL_2( \mathbb{R})/ASL_2( \mathbb{Z})$, that almost every point a curve with some non-degeneracy assumptions is generic for geodesic flow. This implies everywhere locus branched covers torus inside stratum $\mathscr{H}(1,1)$ For these also Kontsevitch-Zorich cocycle, generalizing recent result by Chaika Eskin. As applications, first...

We consider smooth flows preserving a invariant measure, or, equivalently, locally Hamiltonian on compact orientable surfaces and show that, when the genus of surface is two, almost every such flow with two non degenerate isomorphic saddle has singular spectrum. More in general, singularity spectrum holds for special over full measure set interval exchange transformations hyperelliptic permutation (of any number exchanged intervals), under roof symmetric logarithmic singularities. The result...

We consider a symbolic coding of linear trajectories in the regular octagon with opposite sides identified (and more generally 2n-gons). Each infinite trajectory gives cutting sequence corresponding to hit. give an explicit characterization these sequences. The sequences for square are well studied Sturmian which can be analyzed terms continued fraction expansion slope. introduce analogous algorithm we use connect its Our slope substitution operations generate that understood renormalization...