the representation dimension of a selfinjective algebra of euclidean type abstract we prove that the representation dimension of a selfinjective algebra of euclidean type is equal to three and give an explicit construction of the auslander generator of its module category

thermodynamics dimension and the weil petersson metric in this paper we study branched coverings of metrized simplicial trees f t t which arise from polynomial maps f c c with disconnected julia sets we show that the collection of all such trees up to scale forms a contractible space ptd compactifying the moduli space of polynomials of degree d that f records the asymptotic behavior of the multipliers of f and that any meromorphic family of polynomials over can be completed by a unique tree at its central fiber in the cubic case we give a combinatorial enumeration of the trees that arise and show that pt3 is itself a tree resume dans ce travail nous etudions des revetements ramifies d arbres metriques simpliciaux f t t qui sont obtenus a partir d applications polynomiales f c c possedant un ensemble de julia non connexe nous montrons que la collection de tous ces arbres a un facteur d echelle pres forme un espace contractile ptd qui compactifier l espace des modules des polynomes de degre d nous montrons aussi que f enrigistre le comportement asymptotique des multiplicateurs de f et que tout famille meromorphe de polynomes definis sur peut etre completee par un unique arbre comme sa fibre centrale dans le cas cubique nous donnons une enumeration combinatoire des arbres ainsi obtenus et montrons que pt3 est lui meme un arbre

representation dimension and finitely generated cohomology abstract we consider selfinjective artin algebras whose cohomology groups are finitely generated over a central ring of cohomology operators for such an algebra we show that the representation dimension is strictly greater than the maximal complexity occurring among its modules this provides a unified approach to computing lower bounds for the representation dimension of group algebras exterior algebras and artin complete intersections we also obtain new examples of classes of algebras with arbitrarily large representation dimension

representation dimension and quasi hereditary algebras we study auslander s representation dimension of artin algebras which is by definition the minimal projective dimension of coherent functors on modules which are both generators and cogenerators we show the following statements 1 if an artin algebra a is stably hereditary then the representation dimension of a is at most 3 2 if two artin algebras are stably equivalent of morita type then they have the same representation dimension particularly if two self injective algebras are derived equivalent then they have the same representation dimension 3 any incidence algebra of a finite partially ordered set over a field has finite representation dimension moreover we use results on quasi hereditary algebras to show that 4 the auslander algebra of a nakayama algebra has finite representation dimension

the role of polymorphism in the characterisation of complexity by soft types soft type assignment systems sta sta and sta b characterise by means of reduction of terms computation in complexity classes ptime np and pspace respectively all these systems are inspired by linear logic and include polymorphism similar to the one of system f we show that the presence of polymorphism gives the undecidability of typechecking and type inference we also show that reductions in decidable monomorphic versions of these systems also capture the same complexity classes in a way sufficient for the traditional complexity theory the translations we propose show in addition that the monomorphic systems to serve as a programming language require some metalanguage support since the program which operates on data has form and type which depend on the size of the input

diffusion for chaotic plane sections of 3 periodic surfaces we study chaotic plane sections of some particular family of triply periodic surfaces the question about possible behavior of such sections was posed by s p novikov we prove some estimations on the diffusion rate of these sections using the connection between novikov s problem and systems of isometries some natural generalization of interval exchange transformations using thermodynamical formalism we construct an invariant measure for systems of isometries of a special class called the rauzy gasket and investigate the main properties of the lyapunov spectrum of the corresponding suspension flow

representation dimension via tilting theory abstract let be an artin algebra over a commutative artinian ring r and let t be a separating splitting tilting module with endomorphism end t the aim of this paper is to use tilting theory to study the representation dimension of the main result asserts that for an integer n 1 if is n gorenstein of finite cohen macaulay type then r e p d i m n 2 we conclude that if is a n gorenstein artin algebra of finite cohen macaulay type then r e p d i m n 2 this in particular shows that for any artin algebra r e p d i m g l d i m 2

phase transitions for countable markov shifts we study the analyticity of the topological pressure for some one parameter families of potentials on countable markov shifts we relate the non analyticity of the pressure to changes in the recurrence properties of the system we give sufficient conditions for such changes to exist and not to exist we apply these results to the manneville pomeau map and use them to construct examples with different critical behavior

the representation dimension of a selfinjective algebra of wild tilted type abstract we prove that the representation dimension of a selfinjective algebra of wild tilted type is equal to three and give an explicit construction of an auslander generator of its module category we also show that if a connected selfinjective algebra admits an acyclic generalised standard auslander reiten component then its representation dimension is equal to three

symbolic dynamics for surface diffeomorphisms with positive entropy part 1 chains as pseudo orbits 8 2 pesin charts 8 2 1 non uniform hyperbolicity 8 2 2 lyapunov change of coordinates 8 2 3 pesin charts 10 2 4 distortion compensating bounds 11 2 5 nuh f 11 3 overlapping charts 11 3 1 the overlap condition 12 3 2 the form of f in overlapping charts 14 3 3 coarse graining 16 4 e chains and an infinite to one markov extension of f 18 4 1 double charts and e chains 18 4 2 admissible manifolds and the graph transform 20 4 3 a markov extension 23 4 4 the relevant part of the extension 27

equilibrium states of weakly hyperbolic one dimensional maps for holder potentials there is a wealth of results in the literature on the thermodynamic formalism for potentials that are in some sense hyperbolic we show that for a sufficiently regular one dimensional map satisfying a weak hyperbolicity assumption every holder continuous potential is hyperbolic a sample consequence is the absence of phase transitions the pressure function is real analytic on the space of holder continuous functions another consequence is that every holder continuous potential has a unique equilibrium state and that this measure has exponential decay of correlations

thermodynamics of towers of hyperbolic type we introduce a class of continuous maps f of a compact topological space x admitting inducing schemes of hyperbolic type and describe the associated tower constructions we then establish a thermodynamic formalism i e we describe a class of real valued potential functions on x such that f possess a unique equilibrium measure associated to each which minimizes the free energy among the measures that are liftable to the tower we also describe some ergodic properties of equilibrium measures including decay of correlations and the central limit theorem we then study the liftability problem and show that under some additional assumptions on the inducing scheme every measure that charges the base of the tower and has sufficiently large entropy is liftable our results extend those obtained in ps05 ps08 for inducing schemes of expanding types and apply to certain multidimensional maps applications include obtaining the thermodynamic formalism for young s diffeomorphisms the henon family at the first birfucation and the katok map in particular for the katok map we obtain the exponential decay of correlations for equilibrium measures associated to the geometric potentials with 0 t 1

statistical properties of dynamical systems with some hyperbolicity this paper is about the ergodic theory of attractors and conservative dynamical systems with hyperbolic properties on large parts though not necessarily all of their phase spaces the main results are for discrete time systems to put this work into context recall that for axiom a attractors the picture has been fairly complete since the 1970 s see s1 b r2 since then much progress has been made on two fronts there is a general nonuniform theory that deals with properties common to all diffeomorphisms with nonzero lyapunov exponents o p1 ka ly and there are detailed analyses of specific kinds of dynamical systems including for example billiards 1 dimensional and henon type maps s2 bsc hk j bc2 by1 statistical properties such as exponential decay of correlations are not enjoyed by all diffeomorphisms with nonzero lyapunov exponents the goal of this paper is a systematic understanding of these and other properties for a class of dynamical systems larger than axiom a this class will not be defined explicitly but it includes some of the much studied examples by looking at regular returns to sets with good hyperbolic properties one could give systems in this class a simple dynamical representation conditions for the existence of natural invariant measures exponential mixing and central limit theorems are given in terms of the return times these conditions can be checked in concrete situations giving a unified way of proving a number of results some new and some old among the new results are the exponential decay of correlations for a class of scattering billiards and for a positive measure set of henon type maps

radical embeddings and representation dimension given a representation finite algebra b and a subalgebra a of b such that the jacobson radicals of a and b coincide we prove that the representation dimension of a is at most three by a result of igusa and todorov this implies that the finitistic dimension of a is finite

tilting modules and representation dimensions in this paper we compare the representation dimensions of two algebras linked by certain tilting modules our main results can be stated as follows suppose t is a tilting module over a and b enda t then 1 if t is separating and splitting then rep dim a rep dim b 2 if t p 1s is an apr tilting module and the injective dimension of s is at most 2 then rep dim b rep dim a 1

cohomology of twisted tensor products abstract it is well known that the cohomology of a tensor product is essentially the tensor product of the cohomologies we look at twisted tensor products and investigate to which extent this is still true we give an explicit description of the ext algebra of the tensor product of two modules and under certain additional conditions describe an essential part of the hochschild cohomology ring of a twisted tensor product as an application we characterize precisely when the cohomology groups over a quantum complete intersection are finitely generated over the hochschild cohomology ring moreover both for quantum complete intersections and in related cases we obtain a lower bound for the representation dimension of the algebra

representation dimensions of triangular matrix algebras abstract let a be a finite dimensional hereditary algebra over an algebraically closed field k t 2 a a 0 a a be the triangular matrix algebra we prove that rep dim t 2 a is at most three if a is dynkin type and rep dim t 2 a is at most four if a is not dynkin type let a 1 a 0 da a be the duplicated algebra of a let t be a tilting a module and t t p be a tilting a 1 module we show that end a 1 t is representation finite if and only if the full subcategory x a y a f x a mod a y a 1 f t a add a of mod t 2 a is of finite type where is the auslander reiten translation and f t a is the torsion free class of mod a associated with t moreover we also prove that rep dim end a 1 t is at most three if a is dynkin type

specification and towers in shift spaces we show that a shift space on a finite alphabet with a non uniform specification property can be modeled by a strongly positive recurrent countable state markov shift to which every equilibrium state lifts in addition to uniqueness of the equilibrium state this gives strong statistical properties including the bernoulli property exponential decay of correlations central limit theorem and analyticity of pressure which are new even for uniform specification we give applications to shifts of quasi finite type synchronised and coded shifts and factors of beta shifts and s gap shifts

the undecidability of the semi unification problem abstract the semi unification problem sup is a natural generalization of both first order unification and matching the problem arises in various branches of computer science and logic although several special cases of sup are known to be decidable the problem in general has been open for several years we show that sup in general is undecidable by reducing what we call the boundedness problem of turing machines to sup the undecidability of this boundedness problem is established by a technique developed in the mid 1960s to prove related results about turing machines

homology of perfect complexes abstract it is proved that the sum of the loewy lengths of the homology modules of a finite free complex f over a local ring r is bounded below by a number depending only on r this result uncovers in the structure of modules of finite projective dimension obstructions to realizing r as a closed fiber of some flat local homomorphism other applications include as special cases uniform proofs of known results on free actions of elementary abelian groups and of tori on finite cw complexes the arguments use numerical invariants of objects in general triangulated categories introduced here and called levels they allow one to track through changes of triangulated categories homological invariants like projective dimension as well as structural invariants like loewy length an intermediate result sharpens with a new proof the new intersection theorem for commutative algebras over fields under additional hypotheses on the ring r stronger estimates are proved for loewy lengths of modules of finite projective dimension

a simple proof of a theorem of statman abstract in this note we reprove a theorem of statman that deciding the equality of two first order typable terms is not elementary recursive statman 1982 the basic idea of our proof like that of statman s is the henkin quantifier elimination procedure henkin 1963 however our coding is much simpler and easy to understand

the representation dimension of hecke algebras and symmetric groups abstract we establish a lower bound for the representation dimension of all the classical hecke algebras of types a b and d for all the type a algebras and most of the algebras of types b and d we also establish upper bounds moreover we establish bounds for the representation dimension of group algebras of some symmetric groups

an upper bound for the dimension of bounded derived categories abstract let be an artin algebra we give an upper bound for the dimension of the bounded derived category of the category mod of finitely generated right modules in terms of the projective and injective dimensions of certain class of simple right modules as well as the radical layer length of in addition we give an upper bound for the dimension of the singularity category of mod in terms of the radical layer length of

dimensions of triangulated categories via koszul objects lower bounds for the dimension of a triangulated category are provided these bounds are applied to stable derived categories of artin algebras and of commutative complete intersection local rings as a consequence one obtains bounds for the representation dimensions of certain artin algebras

renormalization thermodynamic formalism and quasi crystals in subshifts we examine the thermodynamic formalism for a class of renormalizable dynamical systems which in the symbolic space is generated by the thue morse substitution and in complex dynamics by the feigenbaum coullet tresser map the basic question addressed is whether fixed points v of a renormalization operator mathcal r acting on the space of potentials are such that the pressure function beta mapsto mathcal p beta v exhibits phase transitions this extends the work by baraviera leplaideur and lopes on the manneville pomeau map where such phase transitions were indeed detected in this paper however the attractor of renormalization is a cantor set rather than a single fixed point which admits various classes of fixed points of mathcal r some of which do and some of which do not exhibit phase transitions in particular we show it is possible to reach as a ground state a quasi crystal before temperature zero by freezing a dynamical system

the representation dimension of k x y x2 yn the representation dimension of an artin algebra was defined by m auslander in 1970 the precise value is not known in general and is very hard to compute even for small examples for group algebras it is known in the case of cyclic sylow subgroups for some group algebras in characteristic 2 of rank at least 3 the precise value of the representation dimension follows from recent work of r rouquier there is a gap for group algebras of rank 2 in this paper we show that for all n 0 and any field k the commutative algebras k x y x2 y2 n have representation dimension 3 for the proof we give an explicit inductive construction of a suitable generator cogenerator as a consequence we obtain that the group algebras in characteristic 2 of the groups c2 c2m have representation dimension 3 note that for m 3 these group algebras have wild representation type

spectral gap and transience for ruelle operators on countable markov shifts we find a necessary and sufficient condition for the ruelle operator of a weakly holder continuous potential on a topologically mixing countable markov shift to act with spectral gap on some rich banach space we show that the set of potentials satisfying this condition is open and dense for a variety of topologies we then analyze the complement of this set in a finer topology and show that among the three known obstructions to spectral gap weak positive recurrence null recurrence transience transience is open and dense and null recurrence and weak positive recurrence have empty interior

powers of the phantom ideal the mono epi me exact structure on the morphisms of an exact category a e is introduced and used to prove ideal versions of salce s lemma christensen s ghost lemma and wakamatsu s lemma for an exact category salce s lemma establishes a bijective correspondence i i between the class of special precovering ideals of a e and that of its special preenveloping ideals me extensions of morphisms are used to define an extension i j of ideals christensen s lemma asserts that the class of special precovering respectively special preenveloping ideals is closed under products and extensions and that the bijective correspondence of salce s lemma satisfies ij j i and i j j i wakamatsu s lemma asserts that if a covering ideal i is closed under me extensions then it is a special precovering ideal as an application it is proved that if g is a finite group and is the ideal of phantom morphisms in the category k g mod then n 1 is the object ideal generated by projective modules where n is the nilpotency index of the jacobson radical j if r is a semiprimary ring with jn 0 then n is generated by projective modules for a right coherent ring r over which every cotorsion left r module has a coresolution of length n by pure injective modules n 1 is generated by flat modules

lower bounds for auslander s representation dimension the notion of representation dimension of a finite dimensional algebra has been introduced by auslander he has shown that an algebra is of finite representation type if and only if its representation dimension is at most two he has expected that the representation dimension of a representation infinite algebra should measure how far this algebra is from being representation finite in many cases and thanks to iyama also in general upper bounds for the representation dimension are known however there are non trivial lower bounds only for two similarly constructed series of algebras in this thesis new lower bounds for auslander s representation dimension are constructed and more general methods are developed first we consider group algebras of elementary abelian groups we show that their representation dimension is greater than the rank of the group this leads to a proof of a conjecture of benson concerning the loewy length of blocks of group algebras then a notion of dimension of a module category is introduced this notion enables us to find better lower bounds for the representation dimension in particular it allows us for the first time to determine the value of the representation dimension of non self injective algebras of representation dimension greater than three finally we give a general criterion for finding lower bounds for auslander s representation dimension suppose there is a family of modules such that sufficiently many members of the family have certain d extensions then the representation dimension of the algebra is at least d 2 we illustrate how this criterion is applied on some interesting families of algebras it turns out that the lower bound is often the precise value of the representation dimension

auslander type conditions and cotorsion pairs we study the properties of rings satisfying auslander type conditions if an artin algebra satisfies the auslander condition that is is an gorenstein artin algebra then we construct two kinds of subcategories which form functorially finite cotorsion pairs

the uniqueness of the measure of maximal entropy for geodesic flows on rank 1 manifolds in this paper we prove a conjecture of a katok stating that the geodesic flow on a compact rank 1 manifold admits a uniquely determined invariant measure of maximal entropy this generalizes previous work of r bowen and g margulis as an application we show that the exponential growth rate of the set of singular closed geodesics is strictly smaller than the topological entropy

dimensions of triangulated categories with respect to subcategories abstract this paper introduces a concept of dimension of a triangulated category with respect to a fixed full subcategory for the bounded derived category of an abelian category upper bounds of the dimension with respect to a contravariantly finite subcategory and a resolving subcategory are given our methods not only recover some known results on the dimensions of derived categories in the sense of rouquier but also apply to various commutative and non commutative noetherian rings

unique equilibrium states for flows and homeomorphisms with non uniform structure abstract using an approach due to bowen franco showed that continuous expansive flows with specification have unique equilibrium states for potentials with the bowen property we show that this conclusion remains true using weaker non uniform versions of specification expansivity and the bowen property we also establish a corresponding result for homeomorphisms in the homeomorphism case we obtain the upper bound from the level 2 large deviations principle for the unique equilibrium state the theory presented in this paper provides the basis for an ongoing program to develop the thermodynamic formalism in partially hyperbolic and non uniformly hyperbolic settings

representation dimension of exterior algebras we determine the representation dimension of exterior algebras this provides the first known examples of representation dimension 3 we deduce that the loewy length of the group algebra over f2 of a finite group is strictly bounded below by the 2 rank of the group a conjecture of benson a key tool is the use of the concept of dimension of a triangulated category

natural equilibrium states for multimodal maps this paper is devoted to the study of the thermodynamic formalism for a class of real multimodal maps this class contains but it is larger than collet eckmann for a map in this class we prove existence and uniqueness of equilibrium states for the geometric potentials t log df for the largest possible interval of parameters t we also study the regularity and convexity properties of the pressure function completely characterising the first order phase transitions results concerning the existence of absolutely continuous invariant measures with respect to the lebesgue measure are also obtained

on the finitistic dimension conjecture of artin algebras abstract let a be an artin algebra and e be an idempotent element of a we prove that if a has representation dimension at most three then the finitistic dimension of eae is finite and deduce that if quasi hereditary algebras have representation dimensions at most three then the finitistic dimension conjecture holds

on subcategories closed under predecessors and the representation dimension abstract let be an artin algebra a and c be full subcategories of the category of finitely generated modules consisting of indecomposable modules and closed under predecessors and successors respectively in this paper we relate under various hypotheses the representation dimension of to those of the left support algebra of a and the right support algebra of c our results are then applied to the classes of laura algebras ada algebras and nakayama oriented pullbacks

soft linear logic and polynomial time we present a subsystem of second order linear logic with restricted rules for exponentials so that proofs correspond to polynomial time algorithms and vice versa