hausdorff dimension of the set of nonergodic directions it is known that nonergodic directions in a rational billiard form a subset of the unit circle with hausdorff dimension at most 1 2 explicit examples realizing the dimension 1 2 are constructed using diophantine numbers and continued fractions a lower estimate on the number of primitive lattice points in certain subsets of the plane is used in the construction

singular density of states measure for subshift and quasi periodic schrodinger operators simon s subshift conjecture states that for every aperiodic minimal subshift of verblunsky coefficients the common essential support of the associated measures has zero lebesgue measure we disprove this conjecture in this paper both in the form stated and in the analogous formulation of it for discrete schrodinger operators in addition we prove a weak version of the conjecture in the schrodinger setting namely under some additional assumptions on the subshift we show that the density of states measure a natural measure associated with the operator family and whose topological support is equal to the spectrum is singular we also consider one frequency quasi periodic schrodinger operators with continuous sampling functions and show that generically the density of states measure is singular as well

the cauchy integral analytic capacity and uniform rectifiability several explanations concerning notation terminology and background are in order first notation by 7hi we have denoted the one dimensional hausdorff measure i e length and a z r stands for the closed disc with center z and radius r a curve f is called ad regular that is ahlfors david regular if it satisfies 1 2 with e f since the lower bound is automatic for curves this means that hl r n a z r 0 general sets satisfying 1 2 are called ad regular it is simplest to define the l2 boundedness of the cauchy singular integral operator via the truncated integrals we say that ce is bounded in l2 e without really defining the operator ce itself if there is m ox such that

the set of non uniquely ergodic d iets has hausdorff codimension 1 2 we show that the set of not uniquely ergodic d iets with permutation in the rauzy class of the hyperelliptic permutation has hausdorff dimension d frac 3 2 in the d 1 dimension space of d iets for d ge 5 for d 4 this was shown by athreya chaika and for d in 2 3 the set is known to have dimension d 2 this provides lower bounds on the hausdorff dimension of non weakly mixing iets and with input from al saqban et al exceptional directions for the teichmuller geodesic flow and hausdorff dimension 2017 arxiv 1711 10542 identifies the hausdorff dimension of non weakly mixing iets with permutation d d 1 ldots 2 1 when d is odd

non ergodic z periodic billiards and infinite translation surfaces we give a criterion which proves non ergodicity for certain infinite periodic billiards and directional flows on mathbb 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 mathbb 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 directional billiard flow is not ergodic and has uncountably many ergodic components as another application we show that for any recurrent mathbb 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 skew products which arise as poincare maps of the above systems are associated to non regular mathbb z valued cocycles for interval exchange transformations

mixing for the time changes of heisenberg nilflows we consider reparametrizations of heisenberg nilflows we show that if a heisenberg nilflow is uniquely ergodic all non trivial time changes within a dense subspace of smooth time changes are mixing equivalently in the language of special flows we consider special flows over linear skew shifts over an irrational rotation of the circle without assuming any diophantine condition on the frequency we define a dense class of smooth roof functions for which the corresponding special flows are mixing mixing is produced by a mechanism known as stretching of birkhoff sums the set of mixing timechanges or equivalently roof functions has countable codimension and can be explicitely described in terms of invariant distributions for the nilflow allowing to produce concrete examples of mixing time changes

calderon zygmund kernels and rectifiability in the plane abstract let e c be a borel set with finite length that is 0 h 1 e by a theorem of david and leger the l 2 h 1 e boundedness of the singular integral associated to the cauchy kernel or even to one of its coordinate parts x z 2 y z 2 z x y c implies that e is rectifiable we extend this result to any kernel of the form x 2 n 1 z 2 n z x y c n n we thus provide the first non trivial examples of operators not directly related with the cauchy transform whose l 2 boundedness implies rectifiability

every ergodic transformation is disjoint from almost every interval exchange transformation we show that every transformation is disjoint from almost every interval exchange transformation iet answering a question of bufetov in particular we prove that almost every pair of iets is disjoint it follows that the product of almost every pair is uniquely ergodic a key step in the proof is showing that any sequence of density 1 contains a rigidity sequence for almost every iet strengthening a result of veech

algebraic cycles and higher k theory here g x is the grothendieck group of coherent sheaves on x 13 gr refers to the graded group defined by the y filtration on g x cf kratzer 14 soult 20 and ch x is the chow group of codimension i algebraic cycles defined by fulton 9 the left hand isomorphism is a formal consequence of the existence of a i structure on g x while the existence of r is the central theme of the b f m rr theorem the main purpose of this paper is to define a theory of higher chow groups ch x n n 2 0 so as to obtain isomorphisms

multiscale analysis of 1 rectifiable measures necessary conditions we repurpose tools from the theory of quantitative rectifiability to study the qualitative rectifiability of measures in mathbb r n n ge 2 to each locally finite borel measure mu we associate a function widetilde j 2 mu x which uses a weighted sum to record how closely the mass of mu is concentrated near a line in the triples of dyadic cubes containing x we show that widetilde j 2 mu cdot infty mu a e is a necessary condition for mu to give full mass to a countable family of rectifiable curves this confirms a conjecture of peter jones from 2000 a novelty of this result is that no assumption is made on the upper hausdorff density of the measure thus we are able to analyze general 1 rectifiable measures including measures which are singular with respect to 1 dimensional hausdorff measure

linearization of generalized interval exchange maps a standard interval exchange map is a one to one map of the interval that is locally a translation except at nitely many singularities we dene for such maps in terms of the rauzy veech continuous fraction algorithm a diophantine arithmetical condition called restricted roth type which is almost surely satised in parameter space let t0 be a standard interval exchange map of restricted roth type and let r be an integer 2 we prove that amongst c r 3 deformations of t0 that are c r 3 tangent to t0 at the singularities those that are conjugated to t0 by a c r dieomorphism close to the identity form a c 1 submanifold of codimension g 1 2r 1 s here g is the genus and s is the number of marked points of the translation surface obtained by suspension of t0 both g and s can be computed from the combinatorics of t0

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 poincare 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

an analyst s traveling salesman theorem for sets of dimension larger than one in his 1990 inventiones paper p jones characterized subsets of rectifiable curves in the plane via a multiscale sum of beta numbers these beta numbers are geometric quantities measuring how far a given set deviates from a best fitting line at each scale and location jones result is a quantitative way of saying that a curve is rectifiable if and only if it has a tangent at almost every point moreover computing this square sum for a curve returns the length of the curve up to multiplicative constant k okikiolu extended his result from subsets of the plane to subsets of euclidean space g david and s semmes extended the discussion to include sets of integer dimension larger than one under the assumption of ahlfors regularity and using a variant of jones beta numbers this variant has since been used by others to give structure theorems for rectifiable sets and to give upper bounds for the measure of a set in this paper we give a version of p jones theorem for sets of arbitrary integer dimension lying in euclidean space our main result is a lower bound for the d dimensional hausdorff measure of a set in terms of an analogous sum of beta type numbers we also show an upper bound of this type the combination of these results gives a jones theorem for higher dimensional sets while there is no assumption of ahlfors regularity or of a measure on the underlying set there is an assumption of a lower bound on the hausdorff content we adapt david and semmes version of jones beta numbers by redefining them using a choquet integral allowing them to be defined for arbitrary sets and not just sets of locally finite measure a key tool in the proof is g david and t toro s parametrization of reifenberg flat sets with holes

a family of singular integral operators which control the cauchy transform we study the behaviour of singular integral operators t k t of convolution type on mathbb c associated with the parametric kernels begin aligned k t z frac textsf re z 3 z 4 t cdot frac textsf re z z 2 quad t in mathbb r qquad k infty z frac textsf re z z 2 equiv textsf re frac 1 z quad z in mathbb c setminus 0 end aligned it is shown that for any positive locally finite borel measure with linear growth the corresponding l 2 norm of t k 0 controls the l 2 norm of t k infty and thus of the cauchy transform as a corollary we prove that the l 2 mathcal h 1 lfloor e boundedness of t k t with a fixed t in t 0 0 where t 0 0 is an absolute constant implies that e is rectifiable this is so in spite of the fact that the usual curvature method fails to be applicable in this case moreover as a corollary of our techniques we provide an alternative and simpler proof of the bi lipschitz invariance of the l 2 boundedness of the cauchy transform which is the key ingredient for the bilipschitz invariance of analytic capacity

integral menger curvature for surfaces abstract we develop the concept of integral menger curvature for a large class of nonsmooth surfaces we prove uniform ahlfors regularity and a c 1 a priori bound for surfaces for which this functional is finite in fact it turns out that there is an explicit length scale r 0 which depends only on an upper bound e for the integral menger curvature m p and the integrability exponent p and not on the surface itself below that scale each surface with energy smaller than e looks like a nearly flat disc with the amount of bending controlled by the local m p energy moreover integral menger curvature can be defined a priori for surfaces with self intersections or branch points we prove that a posteriori all such singularities are excluded for surfaces with finite integral menger curvature by means of slicing and iterative arguments we bootstrap the holder exponent up to the optimal one 1 8 p thus establishing a new geometric morrey sobolev imbedding theorem as two of the various possible variational applications we prove the existence of surfaces in given isotopy classes minimizing integral menger curvature with a uniform bound on area and of area minimizing surfaces subjected to a uniform bound on integral menger curvature

rigidity for piecewise smooth homeomorphisms on the circle abstract we find conditions for two piecewise c 2 homeomorphisms f and g of the circle to be c 1 conjugate besides the restrictions on the combinatorics of the maps we assume that the maps have bounded combinatorics and necessary conditions on the one side derivatives of points where f and g are not differentiable we also assume zero mean nonlinearity for f and g

principal values for riesz transforms and rectifiability let e rd with hn e where hn stands for the n dimensional hausdorff measure in this paper we prove that e is n rectifiable if and only if the limit lime 0 y e x y ex y x y n 1dhn y exists hn almost everywhere in e to prove this result we obtain precise estimates from above and from below for the l2 norm of the n dimensional riesz transforms on lipschitz graphs

diophantine properties of iets and general systems quantitative proximality and connectivity three properties of dynamical systems recurrence connectivity and proximality are quantified by introducing and studying the gauges measurable functions corresponding to each of these properties the properties of the proximality gauge are related to the results in the active field of shrinking targets the emphasis in the present paper is on the iets interval exchange transformations mathcal i t mathcal i 0 1 in particular we prove that if an iet t is ergodic relative to the lebesgue measure then the equality a1 liminf n to infty n bigl t n x y bigr 0 holds for a a x y in mathcal i 2 the ergodicity assumption is essential the result does not extend to all minimal iets also the factor n in a1 is optimal e g it cannot be replaced by n ln ln lnn on the other hand for lebesgue almost all 3 iets mathcal i t we prove that for all 0 a2 liminf n to infty n epsilon bigl t n x t n y bigr infty quad text for lebesgue a a x y in mathcal i 2 this should be contrasted with the equality lim inf n t n x t n y 0 for a a x y in mathcal i 2 which holds since mathcal i 2 t times t is ergodic because generic 3 iets mathcal i t are weakly mixing we introduce the notion of entropy of an iet which is related to obtaining estimates of type a2 we also prove that no 3 iet is strongly topologically mixing

on ergodicity of foliations on mathbb z d covers of half translation surfaces and some applications to periodic systems of eaton lenses we consider the geodesic flow defined by periodic eaton lens patterns in the plane and discover ergodic ones among those the ergodicity result on eaton lenses is derived from a result for quadratic differentials on the plane that are pull backs of quadratic differentials on tori ergodicity itself is concluded for mathbb z d covers of quadratic differentials on compact surfaces with vanishing lyapunov exponents

billiards in polygons survey of recent results in the ten or so years since the publication of ref 19 polygonal billiards have remained an active subject of research in the mathematics and physics literature as a result our understanding of the subject although still far from being complete is much better than it was ten years ago this survey attempts to give a broad overview of the dynamics of billiards in polygons with an emphasis on the material that was not in the literature in 1985 this is not a survey of the publications on polygonal billiards and we apologize to authors whose work has not been included the selection of topics has been strongly influenced by the personal taste of the author and by space limitations thus we do not discuss quantum polygonal billiards in particular quantum chaos see e g refs 41 44 and 45 we hope that somebody will write a survey on this important subject in the near future a fascinating aspect of the subject is the interplay between the geometric shape of the billiard table i e a planar curve and the qualitative features of the billiard dynamics hence it is instructive to compare polygonal billiard tables with other classes of billiard tables in particular the smooth strictly convex tables and the dispersing or sinai tables produce strikingly different types of billiard dynamics we recommend refs 46 and 15 for a general overview of the subject and a comparative study of billiard dynamics for various types of billiard tables

simplicity of lyapunov spectra proof of the zorich kontsevich conjecture we prove the zorich kontsevich conjecture that the non trivial lyapunov exponents of the teichmuller ow on any connected component of a stratum of the moduli space of abelian differentials on compact riemann surfaces are all distinct by previous work of zorich and kontsevich this implies the existence of the complete asymptotic lagrangian flag describing the behavior in homology of the vertical foliation in a typical translation surface

flows without wandering points on compact connected surfaces given a compact 2 dimensional manifold m we classify all continuous flows without wandering points on m this classification is performed by finding finitely many pairwise disjoint open invariant subsets u 1 u 2 u n of m such that n i 1 u i m and each u i is either a suspension of an interval exchange transformation or a maximal open cylinder made up of closed trajectories of

smoothness of the beurling transform in lipschitz domains let c be a lipschitz domain and consider the beurling transform of b z lime 0 1 w z w e1 z w 2dm w let 1 1 in this paper we show that if the outward unit normal n on belongs to the besov space bp p 1 p then b is in the sobolev space w p this result is sharp further together with recent results by cruz mateu and orobitg this implies that the beurling transform is bounded in w p if n belongs to bp p 1 p assuming that p 2

quantifying curvelike structures of measures by using l2 jones quantities we study the curvelike structure of special measures on n in a multiscale fashion more precisely we consider the existence and construction of a sufficiently short curve with a sufficiently large measure our main tool is an l2 variant of jones numbers which measure the scaled deviations of the given measure from a best approximating line at different scales and locations the jones function is formed by adding the squares of the l2 jones numbers at different scales and the same location using a special l2 jones function we construct a sufficiently short curve with a sufficiently large measure the length and measure estimates of the underlying curve are expressed in terms of the size of this jones function 2003 wiley periodicals inc

weak mixing directions in non arithmetic veech surfaces we show that the billiard in a regular polygon is weak mixing in almost every invariant surface except in the trivial cases which give rise to lattices in the plane triangle square and hexagon more generally we study the problem of prevalence of weak mixing for the directional flow in an arbitrary non arithmetic veech surface and show that the hausdorff dimension of the set of non weak mixing directions is not full we also provide a necessary condition verified for instance by the veech surface corresponding to the billiard in the pentagon for the set of non weak mixing directions to have positive hausdorff dimension

menger curvature and rectifiability where hi is the 1 dimensional hausdorff measure in rn c x y z is the inverse of the radius of the circumcircle of the triangle x y z that is following the terminology of 6 the menger curvature of the triple x y z a borel set e c rn is said to be purely unrectifiable if for any lipschitz function y r t r 1h l e n r 0 whereas it is said to be rectifiable if there exists a countable family of lipschitz functions yi r e rn such that hi1 e ui yi ir 0 it may be seen from this definition that any 1 set e that is e borel and 0 7 t e oo can be decomposed into two subsets

hybrid quasicrystals transport and localization in products of minimal sets we consider convex combinations of finite valued almost periodic sequences mainly substitution sequences and put them as potentials of one dimensional tight binding models we prove that these sequences are almost periodic we call such combinations hybrid quasicrystals and these studies are related to the minimality under the shift on both coordinates of the product space of the respective minimal hulls we observe a rich variety of behaviors on the quantum dynamical transport ranging from localization to transport

connected components of the moduli spaces of abelian differentials with prescribed singularities consider the moduli space of pairs c where c is a smooth compact complex curve of a given genus and is a holomorphic 1 form on c with a given list of multiplicities of zeroes we describe connected components of this space this classification is important in the study of dynamics of interval exchange transformations and billiards in rational polygons and in the study of geometry of translation surfaces

wasserstein distance and the rectifiability of doubling measures part i let mu be a doubling measure in mathbb r n we investigate quantitative relations between the rectifiability of mu and its distance to flat measures more precisely for x in the support sigma of mu and r 0 we introduce a number alpha x r in 0 1 that measures in terms of a variant of the l 1 wasserstein distance the minimal distance between the restriction of mu to b x r and a multiple of the lebesgue measure on an affine subspace that meets b x r 2 we show that the set of points of sigma where int 0 1 alpha x r dr over r infty can be decomposed into rectifiable pieces of various dimensions we obtain additional control on the pieces and the size of mu when we assume that some carleson measure estimates hold

hitting time and dimension in axiom a systems generic interval exchanges and an application to birkoff sums in this note we prove that for equilibrium states of axiom a systems having positive dimension the time b x needed for a typical point x to enter for the first time in a typical ball b with radius r behaves for small r as b x r d where d is the local dimension of the invariant measure at the center of the ball a similar relation is proved for a full measure set of interval exchanges some applications to birkoff averages of unbounded and not l1 functions are shown

periodic billiard orbits are dense in rational polygons we show that periodic orbits are dense in the phase space for billiards in polygons for which the angle between each pair of sides is a rational multiple of 7r

some remarks concerning an example of a minimal non uniquely ergodic interval exchange transformation on etudie l existence de mesure invariante unique pour une transformation d echange de l intervalle satisfaisant la condition de minimalite

wasserstein distance and the rectifiability of doubling measures part i let mu be a doubling measure in mathbb r n we investigate quantitative relations between the rectifiability of mu and its distance to flat measures more precisely for x in the support sigma of mu and r 0 we introduce a number alpha x r in 0 1 that measures in terms of a variant of the l 1 wasserstein distance the minimal distance between the restriction of mu to b x r and a multiple of the lebesgue measure on an affine subspace that meets b x r 2 we show that the set of points of sigma where int 0 1 alpha x r dr over r infty can be decomposed into rectifiable pieces of various dimensions we obtain additional control on the pieces and the size of mu when we assume that some carleson measure estimates hold

deviation of ergodic averages for substitution dynamical systems with eigenvalues of modulus 1 deviation of ergodic sums is studied for substitution dynamical systems with a matrix that admits eigenvalues of mod ulus 1 the functions we consider are the corresponding eigen functions in theorem 1 1 we prove that the limit inferior of the ergodic sums n x 0 x n 1 n n is bounded for every point x in the phase space in theorem 1 2 we prove existence of limit distributions along certain exponential subsequences of times for substitutions of constant length under additional assumptions we prove that ergodic integrals satisfy the central limit theorem theorem 1 3 theorem 1 9

sharp boundedness and regularizing effects of the integral menger curvature for submanifolds abstract we show that embedded and compact c 1 manifolds have finite integral menger curvature if and only if they are locally graphs of functions belonging to certain sobolev slobodeckij spaces furthermore we prove that for some intermediate energies of integral menger type a similar characterization of objects with finite energy can be given

on rectifiable curves with l p bounds on global curvature self avoidance regularity and minimizing knots we discuss the analytic properties of curves whose global curvature function g 1 is p integrable it turns out that the lp norm mathcal u p gamma rho g gamma 1 l p is an appropriate model for a self avoidance energy interpolating between soft knot energies in form of singular repulsive potentials and hard self obstacles such as a lower bound on the global radius of curvature introduced by gonzalez and maddocks we show in particular that for all p 1 finite mathcal u p energy is necessary and sufficient for w2 p regularity and embeddedness of the curve moreover compactness and lower semicontinuity theorems lead to the existence of mathcal u p minimizing curves in given isotopy classes there are obvious extensions to other variational problems for curves and nonlinearly elastic rods where one can introduce a bound on mathcal u p to preclude self intersections

hyperbolicity and integral expression of the lyapunov exponents for linear cocycles abstract consider in this paper a linear skew product system t w r n w r n t w x t w t w x where t r or z and t w t w is a topological dynamical system on a compact metrizable space w and where t w gl n r satisfies the cocycle condition based on and is continuously differentiable in t if t r we show that semi exponential dichotomy of implies exponential dichotomy precisely if has no lyapunov exponent and is almost uniformly contracting along the stable direction e s w and if dim e s w is constant a e then is almost exponentially dichotomous to prove this we first use liao s spectrum theorem which gives integral expression of the lyapunov exponents and then use the semi uniform ergodic theorem by sturman and stark which allows one to derive uniform estimates from nonuniform ones as a consequence we obtain the open and dense hyperbolicity of eventual gl 2 r cocycles based on a uniquely ergodic endomorphism and of gl 2 r cocycles based on a uniquely ergodic equi continuous endomorphism respectively on the other hand in the sense of c 0 topology we obtain the density of sl 2 r cocycles having positive lyapunov exponent based on a minimal subshift satisfying the boshernitzan condition

bilipschitz maps analytic capacity and the cauchy integral let c c be a bilipschitz map we prove that if e c is compact and e e stand for its analytic and continuous analytic capacity respectively then c 1 e e c e and c 1 e e c e where c depends only on the bilipschitz constant of further we show that if is a radon measure on c and the cauchy transform is bounded on l 2 then the cauchy transform is also bounded on l 2 where is the image measure of by to obtain these results we estimate the curvature of by means of a corona type decomposition

on codings and dynamics of planar piecewise rotations abstract in this paper we investigate codings and dynamics of the piecewise rotations belonging to a sub class of piecewise isometries we show that the irrational set is empty for some piecewise rational rotations under some assumptions while a piecewise irrational rotation has at least one irrational coding by extending the definition of the coding map onto the entire phase space we further prove that the cell corresponding to irrational coding is a single point set for a piecewise irrational rotation

the growth rate for the number of singular and periodic orbits for a polygonal billiard for any simply connected polygon in the plane the number of billiard orbits which begin and end at a vertex grows subexponentially with respect to the length or to the number of reflections this implies that the numbers of isolated periodic orbits and of families of parallel periodic orbits do grow subexponentially the main technical device is a calculation showing that the topological entropy of the poincare map for the billiard flow is equal to zero

weak mixing for interval exchange transformations and translation flows we prove that a typical interval exchange transformation is either weakly mixing or it is an irrational rotation we also conclude that a typical translation flow on a typical translation surface of genus g 2 with prescribed singularity types is weakly mixing

ergodicity of billiard flows and quadratic differentials on considere un systeme billard de 2 objets de masses m 1 et m 2 on montre que pour un ensemble dense de paires m 1 m 2 ce systeme est ergodique

generic continuous spectrum for ergodic schrodinger operators we consider families of discrete schrodinger operators on the line with potentials generated by a homeomorphism on a compact metric space and a continuous sampling function we introduce the concepts of topological and metric repetition property assuming that the underlying dynamical system satisfies one of these repetition properties we show using gordon s lemma that for a generic continuous sampling function the set of elements in the associated family of schrodinger operators that have no eigenvalues is large in a topological or metric sense respectively we present a number of applications particularly to shifts and skew shifts on the torus

closed trajectories for quadratic differentials with an application to billiards on considere les orbites periodiques pour le systeme dynamique d une table polygonale avec des angles multiples rationnels de le billard rationnel

solutions of the cohomological equation for area preserving flows on compact surfaces of higher genus let f be a local homeomorphism of the plane with a fixed point z which is a locally maximal invariant set and which is neither a sink nor a source we prove that there are two integers q 1 and r 1 such that the sequence i fk z of the indices at z of the iterates of f satisfy i fk z 1 rq if k is a multiple of q and i fk z 1 otherwise as a corollary we deduce that there is no minimal homeomorphism on the infinite annulus or more generally on the two dimensional sphere minus a finite set of points we also construct for a local homeomorphism f as above a topological invariant which is a cyclically ordered set with an automorphism on it this allows us in particular to define a rotation number for f rational of denominator q

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 compact connected surface of genus g 2 and phi f t t in mathbb r has the form phi f t x y phi t x y int 0 t f phi s x 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 fancyscript 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

the cohomological equation for roth type interval exchange maps 0 introduction 0 1 interval exchange maps 0 2 the cohomological equation 0 3 summary of the contents 1 the continued fraction algorithm for interval exchange maps 1 1 interval exchange maps 1 2 the continued fraction algorithm 1 3 roth type interval exchange maps 2 the cohomological equation 2 1 the theorem of gottschalk and hedlund 2 2 special birkhoff sums 2 3 estimates for functions of bounded variation 2 4 primitives of functions of bounded variation 3 suspensions of interval exchange maps 3 1 suspension data 3 2 construction of a riemann surface 3 3 compactification of m 3 4 the cohomological equation for higher smoothness 4 proof of full measure for roth type 4 1 the basic operation of the algorithm for suspensions 4 2 the teichmuller flow 4 3 the absolutely continuous invariant measure 4 4 integrability of log z 1 4 5 conditions b and c have full measure 4 6 the main step 4 7 condition a has full measure 4 8 proof of the proposition appendix a roth type conditions in a concrete family of interval exchange maps appendix b a nonuniquely ergodic interval exchange map satisfying condition a

menger curvature and rectifiability in metric spaces we show that for any metric space x the condition where c z1 z2 z3 is the menger curvature of the triple z1 z2 z3 and is the 1 dimensional hausdorff measure on x guarantees that x is rectifiable

limit theorems for translation flows the aim of this paper is to obtain an asymptotic expansion for ergodic integrals of translation flows on flat surfaces of higher genus theorem 1 and to give a limit theorem for these flows theorem 2

a set with finite curvature and projections of zero length a compact subset e of the complex plane is called removable if all bounded analytic functions on its complement are constant or equivalently i f its analytic capacity vanishes the problem of finding a geometric characterization of the removable sets is more than a hundred years old and still not comp letely solved

some calderon zygmund kernels and their relations to wolff capacities and rectifiability we consider the calderon zygmund kernels k alpha n x x i 2n 1 x 2n 1 alpha i 1 d in mathbb r d for 0 alpha le 1 and n in mathbb n we show that on the plane for 0 alpha 1 the capacity associated to the kernels k alpha n is comparable to the riesz capacity c frac 2 3 2 alpha frac 3 2 of non linear potential theory as consequences we deduce the semiadditivity and bilipschitz invariance of this capacity furthermore we show that for any borel set e subset mathbb r d with finite length the l 2 mathcal h 1 lfloor e boundedness of the singular integral associated to k 1 n implies the rectifiability of the set e we thus extend to any ambient dimension results previously known only in the plane

a condition of boshernitzan and uniform convergence in the multiplicative ergodic theorem this paper is concerned with uniform convergence in the multiplicative ergodic theorem on aperiodic subshifts if such a subshift satisfies a certain condition originally introduced by boshernitzan every locally constant sl 2 r valued cocycle is uniform as a consequence the corresponding schrodinger operators exhibit cantor spectrum of lebesgue measure zero an investigation of boshernitzan s condition then shows that these results cover all earlier results of this type and moreover provide various new ones in particular boshernitzan s condition is shown to hold for almost all circle maps and almost all arnoux rauzy subshifts

holder regularity of the solutions of the cohomological equation for roth type interval exchange maps we prove that the solutions of the cohomological equation for roth type interval exchange maps are holder continuous provided that the datum is of class c r with r 1 and belongs to a finite codimension linear subspace

a rational billiard flow is uniquely ergodic in almost every direction on demontre qu un flot de billard rationnel est uniquement ergodique dans presque chaque direction

rectifiable sets and the traveling salesman problem let k c c be a bounded set in this paper we shall give a simple necessary and sufficient condit ion for k to lie in a rectifiable curve we say that a set is a rectifiable curve if it is the image of a finite interval under a lipschitz mapping recall that for a connected set f c c f is a rectifiable curve not necessarily simple if and only if l f where l denotes one dimensional hausdorff measure this classical result follows from the fact that on any finite graph there is a tour which covers the entire graph and which crosses each edge but not necessarily each vertex at most twice if k is a finite set then we are essentially reduced to the classical traveling salesman problem tsp compute the length of the shortest hami l ton ian cycle which hits all points of k this is the same up to a constant multiple as asking for the inf imum of l f where f is a curve k c f such a f is called a spanning tree in tsp theory for infinite sets k we cannot hope in general to have k be a subset of a jordan curve what we should therefore look at is connected sets which conta in k let fmi n be the shortest minimal spanning tree then we cannot possibly solve our problem for sets k of infinite cardinality if we cannot find f i f c o fmin for any finite set k here and throughout the paper c co c1 c o etc denote various universal constants while there are several algorithms for computing l fml these algorithms work for finite graphs satisfying the triangle inequality and do not use the euclidean properties of k see 13 for an excellent discussion of some of these algorithms therefore these methods cannot solve our problem for general infinite k we present a method which is a minor modification of a well known algorithm farthest insert ion see 13 which yields a f with i f c o l fmi the farthest insert ion algorithm has been extensively studied with large numerical calculations on computers and is experimentally good in the sense that the f produced satisfy i f c o l f i for all examples which have

teichmuller curves in moduli space eisenstein series and an application to triangular billiards there exists a teichmuller disc n containing the riemann surface ofy2 x n 1 in the genus n 1 2 teichmuller space such that the stabilizer of n in the mapping class group has a fundamental domain of finite poincare volume in n application is given to an asymptotic formula for the length spectrum of the billiard in isosceles triangles with angles n n n 2 n and to the uniform distribution of infinite billiard trajectories in the same triangles