on the thermodynamic formalism for the gauss map we study the generalized transfer operator open image in new window of the gauss maptx 1 x mod 1 on the unit interval this operator which for 1 is the familiar perron frobenius operator oft can be defined for re 1 2 as a nuclear operator either on the banach spacea d of holomorphic functions over a certain discd or on the hilbert space open image in new window of functions belonging to some hardy class of functions over the half planeh 1 2 the spectra of open image in new window on the two spaces are identical on the space open image in new window is isomorphic to an integral operator open image in new window with kernel the bessel function mathfrak f 2 beta 1 2 sqrt st and hence to some generalized hankel transform this shows that open image in new window has real spectrum for real 1 2 on the spacea d the operator open image in new window can be analytically continued to the entire plane with simple poles at beta beta k 1 k 2 k 0 1 2 and residue the rank 1 operator mathcal n k f tfrac 1 2 1 k f k 0 from this similar analyticity properties for the fredholm determinant open image in new window of open image in new window and hence also for ruelle s zeta function follow another application is to the function zeta m beta sum limits n 1 infty n beta where n denotes the irrational n n n2 4 1 2 2 m extends to a meromorphic function in the plane with the only poles at 1 both with residue 1

ladders and simplicity of derived module categories abstract recollements of derived module categories are investigated using a new technique ladders of recollements which are maximal mutation sequences the position in the ladder is shown to control whether a recollement restricts from unbounded to another level of derived category ladders also turn out to control derived simplicity on all levels an algebra is derived simple if its derived category cannot be deconstructed that is if it is not the middle term of a non trivial recollement whose outer terms are again derived categories of algebras derived simplicity on each level is characterised in terms of heights of ladders these results are complemented by providing new classes of examples of derived simple rings in particular indecomposable commutative rings as well as by a finite dimensional counterexample to the jordan holder problem for derived module categories moreover recollements are used to compute homological and k theoretic invariants

competition interfaces and second class particles the one dimensional nearest neighbor totally asymmetric simple exclusion process can be constructed in the same space as a last passage percolation model in z 2 we show that the trajectory of a second class particle in the exclusion process can be linearly mapped into the competition interface between two growing clusters in the last passage percolation model using technology built up for geodesics in percolation we show that the competition interface converges almost surely to an asymptotic random direction as a consequence we get a new proof for the strong law of large numbers for the second class particle in the rarefaction fan and describe the distribution of the asymptotic angle of the competition interface

superdiffusivity in first passage percolation in standard first passage percolation on bbb z d with d geq 2 the time minimizing paths from a point to a plane at distance l are expected to have transverse fluctuations of order l xi it has been conjectured that xi d geq 1 2 with the inequality strict superdiffusivity at least for low d and with xi 2 2 3 we prove versions of xi d geq 1 2 for all d and xi 2 geq 3 5

true scale invariant random spatial networks some aspects of real world road networks seem to have an approximate scale invariance property motivating study of mathematical models of random networks whose distributions are exactly invariant under euclidean scaling this requires working in the continuum plane so making a precise definition is not trivial we introduce an axiomatization of a class of processes we call scale invariant random spatial networks whose primitives are routes between each pair of points in the plane one concrete model based on minimum time routes in a binary hierarchy of roads with different speed limits has been shown to satisfy the axioms and two other constructions based on poisson line processes and on dynamic proximity graphs are expected also to do so we initiate study of structure theory and summary statistics for general processes in the class many questions arise in this setting via analogies with diverse existing topics from geodesics in first passage percolation to transit node based route finding algorithms

idempotent completion of triangulated categories we show that the idempotent completion of a triangulated category has a natural structure of a triangulated category the idempotent completion of the bounded derived category of an exact category gives the derived category of the idempotent completion in particular the derived category of an idempotent complete exact category is idempotent complete

karoubianness of a triangulated category abstract we prove that triangulated categories with bounded t structures are karoubian consequently for an ext finite abelian category over a commutative noetherian complete local ring its bounded derived category is krull schmidt

critical functions for complex analytic maps critical functions measure the width of the domain of stability around a given fixed point or an invariant circle for complex analytic and area preserving maps the author discusses their dependence on the rotation number of the invariant curves and proposes some new methods to determine them based on the existence of critical points and on some properties of quasiconformal maps by means of the majorant series method some rigorous estimates are given for complex area preserving maps like the semistandard map and the modulated singular map in particular the author makes use of the brjuno function to interpolate critical maps and proves that the convergence of the brjuno function is a necessary and sufficient condition for the existence of an analytic invariant curve of a given rotation number the author also discusses the optimality of the rigorous bounds obtained

intersection products for tensor triangular chow groups abstract we show that under favorable circumstances one can construct an intersection product on the chow groups of a tensor triangulated category t as defined in 5 which generalizes the usual intersection product on a non singular algebraic variety our construction depends on the choice of an algebraic model for t a tensor frobenius pair which has to satisfy a k theoretic regularity condition analogous to the gersten conjecture from algebraic geometry in this situation we are able to prove an analogue of the bloch formula and use it to define an intersection product similar to grayson s construction from 14 we then recover the usual intersection product on a non singular algebraic variety assuming a k theoretic compatibility condition

infinite clusters in percolation models the qualitative nature of infinite clusters in percolation models is investigated the results which apply to both independent and correlated percolation in any dimension concern the number and density of infinite clusters the size of their external surface the value of their total surface to volume ratio and the fluctuations in their density in particular it is shown thatn0 the number of distinct infinite clusters is either 0 1 or and the casen0 which might occur in sufficiently high dimension is analyzed

the brjuno function continuously estimates the size of quadratic siegel disks if a is an irrational number yoccoz defined the brjuno function by a ni 0 a0 a1 i i i an 1 log 1 an where a0 is the fractional part of a and an 1 is the fractional part of 1 an the numbers a such that a i are called the brjuno numbers the quadratic polynomial p a z iu e2i az z2 has an indifferent fixed point at the origin if p a is linearizable we let r a be the conformal radius of the siegel disk and we set r a 0 otherwise yoccoz y proved that a i if and only if r a 0 and that the restriction of a iu a log r a to the set of brjuno numbers is bounded from below by a universal constant in bc2 we proved that it is also bounded from above by a universal constant in fact marmi moussa and yoccoz mmy conjecture that this function extends to r as a hi older function of exponent 1 2 in this article we prove that there is a continuous extension to r

coalescence of geodesics in exactly solvable models of last passage percolation coalescence of semi infinite geodesics remains a central question in planar first passage percolation in this paper we study finer properties of the coalescence structure of finite and semi infinite geodesics for exactly solvable models of last passage percolation consider directed last passage percolation on z2 with independent and identically distributed exponential weights on the vertices fix two points v1 0 0 and v2 0 k2 3 for some k 0 and consider the maximal paths 1 and 2 starting at v1 and v2 respectively to the point n n for n k our object of study is the point of coalescence i e the point v 1 2 with smallest v 1 we establish that the distance to coalescence v 1 scales as k by showing the upper tail bound p v 1 rk r c for some c 0 we also consider the problem of coalescence for semi infinite geodesics for the almost surely unique semi infinite geodesics in the direction 1 1 starting from v3 k2 3 k2 3 and v4 k2 3 k2 3 we establish the optimal tail estimate p v 1 rk r 2 3 for the point of coalescence v this answers a question left open by pimentel ann probab 44 5 3187 3206 2016 who proved the corresponding lower bound

on the relationship between the bruno function and the breakdown invariant tori we study the ratio epsilon c omega exp eta b omega where epsilon c omega is the breakdown threshold function for invariant tori eta is a parameter and b omega is the bruno function which is purely arithmetic i e it only depends by the continued fractions expansion of number omega we consider the standard map as a model and we focus our analysis on the exponential decay of the chaotic regions close to an invariant torus with diophantine rotation frequency our numerical experiments together with some heuristic consideration show that epsilon c omega exp eta b omega is not a continuous function on diophantine numbers omega for all values of eta

non computable impressions of computable external rays of quadratic polynomials we discuss computability of impressions of prime ends of compact sets in particular we construct quadratic julia sets which possess explicitly described non computable impressions

the derived category of an exact category there is some confusion in the literature regarding the derived category of an exact category 8 thomason gives a satisfactory treatment of the bounded derived category in ct 1 11 6 see also appendix a the result is that provided all weakly split epimorphisms in 8 are admissible the bounded derived category may be defined as usual although thomason does not say it the categories d 6 d d may also be defined in the same way the definition of the unbounded derived category is more difficult and is very poorly treated in the literature for instance in bbd this derived category is only defined provided every morphism in 6 admits a kernel see bbd 1 1 41 this is very unsatisfactory since very few exact categories satisfy the condition in this article we will show that d b may be defined whenever 6 is saturated karoubian in thomason s terminology an exact category is saturated if every idempotent splits i e 6 contains all direct summands of its objects for the purpose of comparing with thomason s result the bounded derived category is defined for more 8 s all that is required to define db is that whenever an idempotent e a a factors as a f ba a with fog lb then e is split thomason calls such idempotents weakly split and we will honor his notation we will show here that these constructions are in some sense best possible remark 1 8 for d 6 remarks 1 9 and 1 10 for d d d b db 6 there are two reasons why i wrote this note one is to correct the misconceptions in the literature but more importantly the proof of the key result lemma 1 2 depends on an important characterization of tpaisse subcategories due to rickard and this article is intended to highlight rickard s criterion a full triangulated subcategory y of a triangulated

analytic solutions for a class of differential equation with delays depending on state abstract in this paper we study existence of analytic solutions of a class of the differential equation with state dependent delays x z j 0 k t 1 c t j z x j z t g z in the complex field c by constructing a convergent power series solution y z of an auxiliary equation of the form y z y z j 0 k t 1 c t j y z y j z t g y z with g 0 c 0 invertible analytic solutions of the form y y 1 z for the original equation are obtained

busemann functions and infinite geodesics in two dimensional first passage percolation we study first passage percolation on mathbb z 2 where the edge weights are given by a translation ergodic distribution addressing questions related to existence and coalescence of infinite geodesics some of these were studied in the late 1990s by c newman and collaborators under strong assumptions on the limiting shape and weight distribution in this paper we develop a framework for working with distributional limits of busemann functions and use it to prove forms of newman s results under minimal assumptions for instance we show a form of coalescence of long finite geodesics in any deterministic direction we also introduce a purely directional condition which replaces newman s global curvature condition and whose assumption we show implies the existence of directional geodesics without this condition we prove existence of infinite geodesics which are directed in sectors last we analyze distributional limits of geodesic graphs proving almost sure coalescence and nonexistence of infinite backward paths this result relates to the conjecture of nonexistence of bigeodesics

one hundred years of complex dynamics holomorphic dynamics has been a key area of study for 100 years in this survey we focus on the dynamics of rational functions

bryuno function and the standard map for the standard map the homotopically non trivial invariant curves of rotation number satisfying the bryuno condition are shown to be analytic in the perturbative parameter e provided e is small enough the radius of convergence of the lindstedt series sometimes called critical function of the standard map is studied and the relation with the bryuno function b is derived the quantity log 2 b is proved to be bounded uniformly in

local analytic solutions of a functional differential equation αz βx z x az bx z in this paper we are concerned with the existence of analytic solution of a functional differential equation email protected z x az bx z where a b a b are four complex numbers we first discuss the existence of analytic solutions for some special cases of the above equation then by reducing the equation with the schroder transformation to the another functional equation with proportional delay an existence theorem is established for analytic solutions of the original equation for the constant l given in the schroder transformation we discuss the case 0 l 1 and l on the unit circle s 1 i e l 1 we study l is at resonance i e at a root of the unity and l is near resonance under the brjuno condition

the order of magnitude for moments for certain cotangent sums abstract we settle a question on the rate of growth of the moments of cotangent sums considered by the authors in their previous papers 7 8

negative k theory of derived categories we define negative k groups for exact categories and for derived categories in the framework of frobenius pairs generalizing definitions of bass karoubi carter pedersen weibel and thomason we prove localization and vanishing theorems for these groups devissage for noetherian abelian categories additivity and resolution hold we show that the first negative k group of an abelian category vanishes and that in general negative k groups of a noetherian abelian category vanish our methods yield an explicit non connective delooping of the k theory of exact categories and chain complexes generalizing constructions of wagoner and pedersen weibel extending a theorem of auslander and sherman we discuss the k theory homotopy fiber of e e and its implications for negative k groups in the appendix we replace waldhausen s cylinder functor by a slightly weaker form of non functorial factorization which is still sufficient to prove his approximation and fibration theorems

the first obstructions to enhancing a triangulated category in this paper we relate triangulated category structures to the cohomology of small categories and define initial obstructions to the existence of an algebraic or topological enhancement we show that these obstructions do not vanish in an example of triangulated category without models we also obtain cohomological characterizations of pre triangulated dg a infinity and spectral categories

explicit estimates of sums related to the nyman beurling criterion for the riemann hypothesis abstract we give an estimate for sums appearing in the nyman beurling criterion for the riemann hypothesis these sums contain the mobius function and are related to the imaginary part of the estermann zeta function the estimate is remarkably sharp in comparison to other sums containing the mobius function the bound is smaller than the trivial bound essentially the number of terms by a fixed power of that number the exponent is made explicit the methods intensively use tools from the theory of continued fractions and from the theory of fourier series

generalized serre duality abstract we introduce a notion of generalized serre duality on a hom finite krull schmidt triangulated category t this duality induces the generalized serre functor on t which is a linear triangle equivalence between two thick triangulated subcategories of t moreover the domain of the generalized serre functor is the smallest additive subcategory of t containing all the indecomposable objects which appear as the third term of an auslander reiten triangle in t dually the range of the generalized serre functor is the smallest additive subcategory of t containing all the indecomposable objects which appear as the first term of an auslander reiten triangle in t we compute explicitly the generalized serre duality on the bounded derived categories of artin algebras and of certain noncommutative projective schemes in the sense of artin and zhang we obtain a characterization of gorenstein algebras an artin algebra a is gorenstein if and only if the bounded homotopy category of finitely generated projective a modules has serre duality in the sense of bondal and kapranov

noncommutative mirror symmetry for punctured surfaces in 2013 abouzaid auroux efimov katzarkov and orlov showed that the wrapped fukaya categories of punctured spheres and finite unbranched covers of punctured spheres are derived equivalent to the categories of singularities of a superpotential on certain crepant resolutions of toric 3 dimensional singularities we generalize this result to other punctured riemann surfaces and reformulate it in terms of certain noncommutative algebras coming from dimer models in particular given any consistent dimer model we can look at a subcategory of noncommutative matrix factorizations and show that this category is mathtt a infty isomorphic to a subcategory of the wrapped fukaya category of a punctured riemann surface the connection between the dimer model and the punctured riemann surface then has a nice interpretation in terms of a duality on dimer models

descent in triangulated categories we establish effective descent for faithful ring objects in tensor triangulated categories more generally we discuss descent for monads in triangulated categories without tensor where the answer is more subtle

geodesics and spanning trees for euclidean first passage percolation the metric d q q on the set q of particle locations of a homogeneous poisson process on r d defined as the infimum of i q i q i 1 1 over sequences in q starting with q and ending with q where denotes euclidean distance has nontrivial geodesics when a 1 the cases 1 are the euclidean first passage percolation fpp models introduced earlier by the authors while the geodesics in the case are exactly the paths from the euclidean minimal spanning trees forests of aldous and steele we compare and contrast results and conjectures for these two situations new results for 1 and any d include inequalities on the fluctuation exponents for the metric x 1 2 and for the geodesies 3 4 in strong enough versions to yield conclusions not yet obtained for lattice fpp almost surely every semiinfinite geodesic has an asymptotic direction and every direction has a semiinfinite geodesic from every q for d 2 and 2 further results follow concerning spanning trees of semiinfinite geodesics and related random surfaces

density and uniqueness in percolation two results on site percolation on thed dimensional lattice d 1 arbitrary are presented in the first theorem we show that for stationary underlying probability measures each infinite cluster has a well defined density with probability one the second theorem states that if in addition the probability measure satisfies the finite energy condition of newman and schulman then there can be at most one infinite cluster with probability one the simple arguments extend to a broad class of finite dimensional models including bond percolation and regular lattices

existence of analytic solutions of an iterative functional equation abstract this paper is concern analytic solutions of an iterative functional equation of the form f p z q f z h f z z c byconstructing a convergent power series solution of an auxiliary equation g 2 z p z h g z p z z c analytic solutions of the original equation are obtained we discuss not only these appeared in the auxiliary equation at the hyperbolic case 0

multifractal analysis of the brjuno function the brjuno function b is a 1 periodic nowhere locally bounded function introduced by yoccoz because it encapsulates a key information concerning analytic small divisor problems in dimension 1 we show that t p alpha regularity introduced by calderon and zygmund is the only one which is relevant in order to unfold the pointwise regularity properties of b we determine its t p alpha regularity at every point and show that it is directly related to the irrationality exponent tau x its p exponent at x is exactly 1 tau x this new example of multifractal function puts into light a new link between dynamical systems and fractal geometry finally we also determine the holder exponent of a primitive of b

a lower bound in an approximation problem involving the zeros of the riemann zeta function abstract we slightly improve the lower bound of baez duarte balazard landreau and saias in the nyman beurling formulation of the riemann hypothesis as an approximation problem we construct hilbert space vectors which could prove useful in the context of the so called hilbert polya idea

geodesics and the competition interface for the corner growth model we study the directed last passage percolation model on the planar integer lattice with nearest neighbor steps and general i i d weights on the vertices outside the class of exactly solvable models in georgiou et al probab theory relat fields 2016 doi 10 1007 s00440 016 0729 x we constructed stationary cocycles and busemann functions for this model using these objects we prove new results on the competition interface on existence uniqueness and coalescence of directional semi infinite geodesics and on nonexistence of doubly infinite geodesics

the brjuno functions and their regularity properties we show that various possible versions of the brjuno function based on different kinds of continued fraction developments are all equivalent and we study their regularity l p bmo and holder properties through a systematic analysis of the functional equation which they fulfill

stationary cocycles and busemann functions for the corner growth model we study the directed last passage percolation model on the planar square lattice with nearest neighbor steps and general i i d weights on the vertices outside of the class of exactly solvable models stationary cocycles are constructed for this percolation model from queueing fixed points these cocycles serve as boundary conditions for stationary last passage percolation solve variational formulas that characterize limit shapes and yield existence of busemann functions in directions where the shape has some regularity in a sequel to this paper the cocycles are used to prove results about semi infinite geodesics and the competition interface

on computational complexity of siegel julia sets it has been previously shown by two of the authors that some polynomial julia sets are algorithmically impossible to draw with arbitrary magnification on the other hand for a large class of examples the problem of drawing a picture has polynomial complexity in this paper we demonstrate the existence of computable quadratic julia sets whose computational complexity is arbitrarily high

complex brjuno functions 1 1 the real brjuno function let r q and let pn qn n 0 be the sequence of the convergents of its continued fraction expansion a brjuno number is an irrational number such that n 0 log qn 1 qn the importance of brjuno numbers comes from the study of one dimensional analytic small divisors problems in the case of germs of holomorphic diffeomorphisms of one complex variable with an indifferent fixed point extending a previous result of siegel s brjuno proved br1 br2 that all germs with linear part e are linearizable if is a brjuno number conversely the third author proved that this condition is also necessary yo1 similar results hold for the local conjugacy of analytic diffeomorphisms of the circle kh yo2 yo3 and for some area preserving maps ma da1 including the standard family da2 bg1 bg2 the set of brjuno numbers is invariant under the action of the modular group pgl 2 z and it can be characterized as the set where the brjuno function b r q r is finite this arithmetical function is z periodic and satisfies the remarkable functional equation

non computable julia sets polynomial julia sets have emerged as the most studied examples of fractal sets generated by a dynamical system apart from the beautiful mathematics one of the reasons for their popularity is the beauty of the computer generated images of such sets the algorithms used to draw these pictures vary the most naive work by iterating the center of a pixel to determine if it lies in the julia set milnor s distance estimator algorithm mil uses classical complex analysis to give a one pixel estimate of the julia set this algorithm and its modifications work quite well for many examples but it is well known that in some particular cases computation time will grow very rapidly with increase of the resolution moreover there are examples even in the family of quadratic polynomials when no satisfactory pictures of the julia set exist in this paper we study computability properties of julia sets of quadratic polynomials under the definition we use a set is computable if roughly speaking its image can be generated by a computer with an arbitrary precision under this notion of computability we show

continued fractions and brjuno functions abstract for 0 1 given we consider the modified continued fraction expansion of the real number x defined by x a 0 e 0 x 0 a 0 z and x n 1 1 a n e n x n a n n for n 0 where 1 e n x n e n 1 for n 0 with xn 0 the usual gaussian case is 1 whereas 1 2 is the continued fraction to the nearest integer the brjuno function b x is then defined by b x ln x0 n 1 x0x1 xn 1ln xn these functions were introduced by yoccoz in the 1 and 1 2 cases in his work on the holomorphic conjugacy to a rotation of an analytic map with an indifferent fixed point we will review some properties of these functions namely all these functions are 1 periodic and belong to l ioc p r for 1 p t in this communication we will mainly report on some of the technical tools related to the continued fraction expansions required by the above mentioned results these results deal with the growth of the denominator of the reduced fractions pn qn of the above continued fraction expansion which gives the maximal error rate of approximation the relation between these continued fraction and the usual gaussian case and finally the invariant density generalising the classical result of gauss for the usual case

separability and triangulated categories abstract we prove that the category of modules over a separable ring object in a tensor triangulated category admits a unique structure of triangulated category which is compatible with the original one this applies in particular to etale algebras more generally we do this for exact separable monads

picard groups in triangular geometry and applications to modular representation theory for a tensor triangulated p category k with spectrum spc k we construct an injective group homomorphism h 1 spc k m 1 p pic k 1 p where pic k is the group of invertible objects of k in modular representation theory we prove that this homomorphism induces a rational isomorphism between the picard group of the projective support variety and the group of endotrivial representations

local invertible analytic solutions for an iterative differential equation related to a discrete derivatives sequence abstract in this paper we are concerned with the existence of analytic solutions of a class of iterative differential equation f z 1 k f 1 z a 1 f 2 z a 2 f n z a n in the complex field c where k c 0 a i r f i z denotes ith iterate of f z i 1 2 n the above equation is closely related to a discrete derivatives sequence f m see y f s petermann jean luc remy ilan vardi discrete derivative of sequences adv in appl math 27 2001 562 584 we first give the existence of analytic solutions of the form of power functions for such an equation then by constructing a convergent power series solution y z of an auxiliary equation of the form x z k x z x z a 1 x 2 z a 2 x n z a n invertible analytic solutions of the form f z x x 1 z for the original equation are obtained we discuss not only the constant at resonance i e at a root of the unity but also those near resonance near a root of the unity under the brjuno condition

on the differential graded eilenberg moore construction abstract the eilenberg moore construction for modules over a differential graded monad is used to study a question of balmer regarding existence of an exact adjoint pair representing an exact monad a bousfield like localization for differential graded categories is realized as a special case of this construction using drinfeld quotients as applications we study some example coming from g equivariant triangulated categories and twisted derived categories

absence of geodesics in first passage percolation on a half plane an h geodesic is a doubly infinite path which locally minimizes the passage time in the i i d first passage percolation model on a half plane h under the assumption that the bond passage times are continuously distributed with a finite mean we prove that with probability 1 h geodesics do not exist as a corollary we show that with probability 1 any geodesic in the analogous model on the whole plane z 2 has to intersect all straight lines with rational slopes

a proof of the marmi moussa yoccoz conjecture for rotation numbers of high type marmi moussa and yoccoz conjectured that some error function upsilon related to the approximation of the size of siegel disks by some arithmetic function of the rotation number theta is a holder continuous function of theta with exponent 1 2 using the renormalization invariant class of inou and shishikura we prove this conjecture for the restriction of upsilon to a class of high type numbers

the relative singularity category of a non commutative resolution of singularities abstract in this article we study a triangulated category associated with a non commutative resolution of singularities in particular we give a complete description of this category in the case of a curve with nodal singularities classifying its indecomposable objects and computing its auslander reiten quiver and k group

euclidean models of first passage percolation we introduce a new family of first passage percolation fpp models in the context of poisson voronoi tesselations of d compared to standard fpp on d these models have some technical complications but also have the advantage of statistical isotropy we prove two almost sure results a shape theorem where isotropy implies an exact euclidean ball for the asymptotic shape and nonexistence of certain doubly infinite geodesics where isotropy yields a stronger result than in standard fpp

first passage percolation on mathbb z 2 a simulation study first passage percolation on mathbb z 2 is a model for describing the spread of an infection on the sites of the square lattice the infection is spread via nearest neighbor sites and the time dynamic is specified by random passage times attached to the edges in this paper the speed of the growth and the shape of the infected set is studied by aid of large scale computer simulations with focus on continuous passage time distributions it is found that the most important quantity for determining the value of the time constant which indicates the inverse asymptotic speed of the growth is mathbf e min tau 1 ldots tau 4 where tau 1 ldots tau 4 are i i d passage time variables the relation is linear for a large class of passage time distributions furthermore the directional time constants are seen to be increasing when moving from the axis towards the diagonal so that the limiting shape is contained in a circle with radius defined by the speed along the axes the shape comes closer to the circle for distributions with larger variability

presheaves of triangulated categories and reconstruction of schemes to any triangulated category with tensor product k otimes we associate a topological space rm spc k otimes by means of thick subcategories of k a la hopkins neeman thomason moreover to each open subset u of this space spc k otimes we associate a triangulated category cal k u producing what could be thought of as a presheaf of triangulated categories applying this to the derived category k otimes rm d rm perf x otimes l of perfect complexes on a noetherian scheme x the topological space rm spc k otimes turns out to be the underlying topological space of x moreover for each open u subset x the category cal k u is naturally equivalent to rm d rm perf u as an application we give a method to reconstruct any reduced noetherian scheme x from its derived category of perfect complexes rm d rm perf x considering the latter as a tensor triangulated category with otimes l

constructing locally connected non computable julia sets a locally connected quadratic siegel julia set has a simple explicit topological model such a set is computable if there exists an algorithm to draw it on a computer screen with an arbitrary resolution we constructively produce parameter values for siegel quadratics for which the julia sets are non computable yet locally connected

a normal form theorem for brjuno skew systems through renormalization abstract we develop a dynamical renormalization method for the problem of local reducibility of analytic linear quasi periodic skew product flows on t 2 sl 2 r this approach is based on the continued fraction expansion of the linear base flow and deals with small divisors by turning them into large divisors we use this method to give a new proof of a normal form theorem for a brjuno base flow

stationary coalescing walks on the lattice we consider translation invariant measures on families of nearest neighbor semi infinite walks on the integer lattice we assume that once walks meet they coalesce in 2d we classify the collective behavior of these walks under mild assumptions they either coalesce almost surely or form bi infinite trajectories bi infinite trajectories form measure preserving dynamical systems have a common asymptotic direction in 2d and possess other nice properties we use our theory to classify the behavior of compatible families of semi infinite geodesics in stationary first and last passage percolation we also partially answer a question raised by c hoffman about the limiting empirical measure of weights seen by geodesics we construct several examples our main example is a standard first passage percolation model where geodesics coalesce almost surely but have no asymptotic direction or average weight

variational formulas and cocycle solutions for directed polymer and percolation models we discuss variational formulas for the law of large numbers limits of certain models of motion in a random medium namely the limiting time constant for last passage percolation and the limiting free energy for directed polymers the results are valid for models in arbitrary dimension steps of the admissible paths can be general the environment process is ergodic under spatial translations and the potential accumulated along a path can depend on the environment and the next step of the path the variational formulas come in two types one minimizes over gradient like cocycles and another one maximizes over invariant measures on the space of environments and paths minimizing cocycles can be obtained from busemann functions when these can be proved to exist the results are illustrated through 1 1 dimensional exactly solvable examples periodic examples and polymers in weak disorder

geodesics in two dimensional first passage percolation we consider standard first passage percolation on z 2 geodesics are nearest neighbor paths in z 2 each of whose segments is time minimizing we prove part of the conjecture that doubly infinite geodesics do not exist our main tool is a result of independent interest about the coalescing of semi infinite geodesics