shape and morse theory of attractors we study the shape in the sense of borsuk of attractors of continuous semi dynamical systems on general metric spaces we show in particular that the natural inclusion of the global attractor into the state space is a shape equivalence this and other results of the paper are used to develop an elementary morse theory of an attractor 2000 john wiley sons inc

fractal basin boundaries abstract basin boundaries for dynamical systems can be either smooth or fractal this paper investigates fractal basin boundaries one practical consequence of such boundaries is that they can lead to great difficulty in predicting to which attractor a system eventually goes the structure of fractal basin boundaries can be classified as being either locally connected or locally disconnected examples and discussion of both types of structures are given and it appears that fractal basin boundaries should be common in typical dynamical systems lyapunov numbers and the dimension for the measure generated by inverse orbits are also discussed

strange nonchaotic attractors with wada basins a b s t r a c t we demonstrate strange nonchaotic attractors with wada basins snaswb and verify the abundance of snaswb in a quasiperiodically forced holmes map we identify the routes to the creation of the snaswb in a two parameter space the snaswb are characterized by the maximal lyapunov exponent by the estimation of the phase sensitivity exponent and by the singular continuous spectra we observe that the snas basins are totally wada basins in a large range of parameters the topological structures of the snas wada basins are distinguished by the basin cell method we investigate the underlying mechanism for the abundance of snaswb which is responsible for different types of basin cell in the absence of forcing this suggests that the snas cannot be predicted reliably for specific initial conditions these snaswb can thus be expected to occur more commonly in dynamical systems

unstable dimension variability a source of nonhyperbolicity in chaotic systems the hyperbolicity or nonhyperbolicity of a chaotic set has profound implications for the dynamics on the set a familiar mechanism causing nonhyperbolicity is the tangency of the stable and unstable manifolds at points on the chaotic set here we investigate a different mechanism that can lead to nonhyperbolicity in typical invertible respectively noninvertible maps of dimension 3 respectively 2 and higher in particular we investigate a situation first considered by abraham and smale in 1970 for different purposes in which the dimension of the unstable and stable tangent spaces are not constant over the chaotic set we call this unstable dimension variability a simple two dimensional map that displays behavior typical of this phenomenon is presented and analyzed

intertwined basins of attraction of dynamical systems in this paper we investigate a kind of intertwining phenomenon and give a new definition of intertwined attractors of the dynamical systems and get some related results

wada basin boundaries and basin cells in dynamical systems examples are common in which two or more attractors coexist and in such cases the basin boundary is nonempty we consider a two dimensional diffeomorphism f that is f is an invertible map and both f and its inverse are differentiable with continuous derivatives which has at least three basins fractal basin boundaries contain infinitely many periodic points generally only finitely many of these periodic points are outermost on the basin boundary that is accessible from a basin for many systems all accessible points lie on stable manifolds of periodic points a point x on the basic boundary is a wada point if every open neighborhood of x has a nonempty intersection with at least three different basins we call the boundary of a basin a wada basin boundary if all its points are wada points our main goal is to have definitions and hypotheses for wada basin boundaries that can be verified by computer the basic notion basin cell will play a fundamental role in our results for numerical verifications assuming each accessible point on the boundary of a basin b is on the stable manifold of some periodic orbit we show that partial derivative b over bar is a wada basin boundary if the unstable manifold of each of its accessible periodic orbits intersects at least three basins in addition we find conditions for basins b 1 b 2 b n n greater than or equal to 3 to satisfy partial derivative b over bar 1 partial derivative b over bar 2 partial derivative b over bar n our results provide numerically verifiable conditions guaranteeing that the boundary of a basin is a wada basin boundary our examples make use of an existing numerical procedure for finding the accessible periodic points on the basin boundary and another procedure for plotting stable and unstable manifolds to verify the existence of wada basin boundaries

ergodic averages of commuting transformations with distinct degree polynomial iterates we prove mean convergence as n to infty for the multiple ergodic averages frac 1 n sum n 1 n f 1 t 1 p 1 n x cdot ldots cdot f ell t ell p ell n x where p 1 ldots p ell are integer polynomials with distinct degrees and t 1 ldots t ell are invertible commuting measure preserving transformations acting on the same probability space this establishes several cases of a conjecture of bergelson and leibman that complement the case of linear polynomials recently established by tao furthermore we show that unlike the case of linear polynomials for polynomials of distinct degrees the corresponding characteristic factors are mixtures of inverse limits of nilsystems we use this particular structure together with some equidistribution results on nilmanifolds to give an application to multiple recurrence and a corresponding one to combinatorics

powers of sequences and recurrence we study recurrence and multiple recurrence properties along the k th powers of a given set of integers we show that the property of recurrence for some given values of k does not give any constraint on the recurrence for the other powers this is motivated by similar results in number theory concerning additive basis of natural numbers moreover motivated by a result of kamae and mendes france that links single recurrence with uniform distribution properties of sequences we look for an analogous result dealing with higher order recurrence and make a related conjecture

the classification of topologically expansive lorenz maps we show that topologically expansive lorenz maps can be described up to topological conjugacy by their kneading invariants we also give a simple condition on pairs of symbol sequences which is satisfied if and only if that pair of sequences is the kneading invariant for some topologically expansive lorenz map a simple extension of the theorems to the case of expansive maps of the interval with multiple discontinuities is described

nonconventional ergodic averages and nilmanifolds we study the l2 convergence of two types of ergodic averages the first is the average of a product of functions evaluated at return times along arithmetic progressions such as the expressions appearing in furstenberg fs proof of szemer ledi fs theorem the second average is taken along cubes whose sizes tend to for each average we show that it is sufficient to prove the convergence for special systems the characteristic factors we build these factors in a general way independent of the type of the average to each of these factors we associate a natural group of transformations and give them the structure of a nilmanifold from the second convergence result we derive a combinatorial interpretation for the arithmetic structure inside a set of integers of positive upper density

a multidimensional szemeredi theorem for hardy sequences of different growth abstract we prove a variant of the multidimensional polynomial szemeredi theorem of bergelson and leibman where one replaces polynomial sequences with other sparse sequences defined by functions that belong to some hardy field and satisfy certain growth conditions we do this by studying the limiting behavior of the corresponding multiple ergodic averages and obtaining a simple limit formula a consequence of this formula in topological dynamics shows denseness of certain orbits when the iterates are restricted to suitably chosen sparse subsequences another consequence is that every syndetic set of integers contains certain nonshift invariant patterns and every finite coloring of n with each color class a syndetic set contains certain polychromatic patterns results very particular to our non polynomial setup

strange attractors of uniform flows on considere des attracteurs orbitalement stables de ces flots sur le tore solide ouvert d 2 s 1 qui ont une vitesse dans la direction s 1 on trouve qu un tel attracteur est la frontiere d une suite strictement emboitee de tores solides ouverts positivement invariants

wada dye boundaries in open hydrodynamical flows abstract dyes of different colors advected by two dimensional flows which are asymptotically simple can form a fractal boundary that coincides with a chaotic saddle s unstable manifold we show that such dye boundaries can have the wada property every boundary point of a given color on this fractal set is on the boundary of at least two other colors the condition for this is the nonempty intersection of the saddle s stable manifold with at least three differently colored domains in the asymptotic inflow region

unpredictable behavior in the duffing oscillator wada basins this paper describes some numerical experiments giving evidence of wada basin boundaries for the duffing oscillator we suggest some mechanisms by which this fractal property of the boundaries appears and discuss the difficulties that the wada property presents for predicting the behavior of dynamical systems

linear equations in primes consider a system of nonconstant affine linear forms 1 t d no two of which are linearly dependent let n be a large integer and let k n n d be convex a generalisation of a famous and difficult open conjecture of hardy and littlewood predicts an asymptotic as n for the number of integer points n d k for which the integers 1 n t n are simultaneously prime this implies many other well known conjectures such as the twin prime conjecture and the weak goldbach conjecture it also allows one to count the number of solutions in a convex range to any simultaneous linear system of equations in which all unknowns are required to be prime in this paper we conditionally verify this asymptotic under the assumption that no two of the affine linear forms 1 t are affinely related this excludes the important binary cases such as the twin prime or goldbach conjectures but does allow one to count nondegenerate configurations such as arithmetic progressions our result assumes two families of conjectures which we term the inverse gowers norm conjecture gi s and the mobius and nilsequences conjecture mn s where s 1 2 is the complexity of the system and measures the extent to which the forms i depend on each other the case s 0 is somewhat degenerate and follows from the prime number theorem in aps roughly speaking the inverse gowers norm conjecture gi s asserts the gowers u s 1 norm of a function f n 1 1 is large if and only if f correlates with an s step nilsequence while the mobius and nilsequences conjecture mn s asserts that the mobius function is strongly asymptotically orthogonal to s step nilsequences of a fixed complexity these conjectures have long been known to be true for s 1 essentially by work of hardy littlewood and vinogradov and were established for s 2 in two papers of the authors thus our results in the case of complexity s 2 are unconditional in particular we can obtain the expected asymptotics for the number of 4 term progressions p 1 p 2 p 3 p 4 n of primes and more generally for any nondegenerate problem involving two linear equations in four prime unknowns

pseudocircles in dynamical systems we construct an example of a coo map on a 3 manifold which has an invariant set with an uncountable number of components each of which is a pseudocircle furthermore any map which is sufficiently close in the c1metric to the constructed map has a similar set

contextual insertions deletions and computability we investigate two generalizations of insertion and deletion of words that have recently become of interest in the context of molecular computing given a pair of words x y called a context the x y contextual insertion of a wordvinto a worduis performed as follows for each occurrence ofxyas a subword inu we include in the result of the contextual insertion the words obtained by insertingvintou betweenxandy the x y contextual deletion operation is defined in a similar way we study closure properties of the chomsky families under the defined operations contextual ins closed and del closed languages and decidability of existence of solutions to equations involving these operations moreover we prove that every turing machine can be simulated by a system based entirely on contextual insertions and deletions

multiple recurrence and nilsequences aiming at a simultaneous extension of khintchine s and furstenberg s recurrence theorems we address the question if for a measure preserving system x mathcal x mu t and a set a in mathcal x of positive measure the set of integers n such that mu a cap t n a cap t 2n a cap ldots cap t kn a mu a k 1 epsilon is syndetic the size of this set surprisingly enough depends on the length k 1 of the arithmetic progression under consideration in an ergodic system for k 2 and k 3 this set is syndetic while for k 4 it is not the main tool is a decomposition result for the multicorrelation sequence int f x f t n x f t 2n x ldots f t kn x d mu x where k and n are positive integers and f is a bounded measurable function we also derive combinatorial consequences of these results for example showing that for a set of integers e with upper banach density d e 0 and for all e 0 the set big n in mathbb z colon d big e cap e n cap e 2n cap e 3n big d e 4 epsilon big is syndetic

chaos periodicity and snakelike continua the results of this paper relate the dynamics of a continuous map f of the interval and the topology of the inverse limit space with bonding map f these inverse limit spaces have been studied by many authors and are examples of what bing has called snakelike continua roughly speaking we show that when the dynamics of f are complicated the inverse limit space contains indecomposable subcontinua we also establish a partial converse

on the global structure of invariant regions of flows with asymptotically stable attractors several authors have pointed out the usefulness of shape theory which was introduced by k borsuk in 1968 as a tool in the study of dynamical systems in particular bogatyi and gutsu 4 garay 10 gunther and segal 11 hastings 12 robbin and salamon 16 rogers 17 and tezer 24 have obtained interesting results in dynamical systems using shape theoretical techniques one of the authors of the present article has given another connection between shape and topological dynamics 20 based on the approach to shape developed in 18 and 19 see also 21 for applications of shape theory to the study of uniform attractors most of the mentioned papers study properties of flows although 16 and 24 are devoted to discrete dynamical systems in particular a shape conley index is constructed in 16 for discrete dynamical systems and some shape theoretical ideas are used in 24 to study properties of topological conjugacy of shifts and homotopical shift equivalence for maps of polyhedra in the papers 4 10 11 12 20 21 some topological properties of the attractors of flows are studied for instance hastings established in 12 a higher dimensional poincare bendixson theorem proving the existence of attractors of certain flows in the interior of submanifolds of the euclidean space and comparing the shape of the attractor with that of the manifold other properties studied in the above papers are related to strong cellularity of the global attractors 10 and polyhedral shape of asymptotically stable compacta 4 11 20 a remarkable fact proved by

intertwined basins of attraction generated by the stable manifold of a saddle point this article deals with the intertwined basins of attraction we give a necessary and sufficient condition for the existence of intertwined basins of attraction

realizing rotation numbers on annular continua an annular continuum is a compact connected set k which separates a closed annulus a into exactly two connected components one containing each boundary component the topology of such continua can be very intricate for instance non locally connected we adapt a result proved by handel in the case where k a showing that if k is an invariant annular continuum of a homeomorphism of a isotopic to the identity then the rotation set in k is closed moreover every element of the rotation set is realized by an ergodic measure supported in k and by a periodic orbit if the rotation number is rational and most elements are realized by a compact invariant set our second result shows that if the continuum k is minimal with the property of being annular what we call a circloid then every rational number between the extrema of the rotation set in k is realized by a periodic orbit in k as a consequence the rotation set is a closed interval and every number in this interval rational or not is realized by an orbit moreover by an ergodic measure in k this improves a previous result of barge and gillette

set polynomials and polynomial extension of the hales jewett theorem an abstract hales jewett type extension of the polynomial van der waerden theorem bl is established theorem let r d q n there exists n n such that for any r coloring of the set of subsets of v 1 n d 1 q there exist a set a v and a nonempty set 1 n such that a d 1 q and the subsets a a d 1 a d 2 a d q are all of the same color this polynomial hales jewett theorem contains refinements of many combinatorial facts as special cases the proof is achieved by introducing and developing the apparatus of set polynomials polynomials whose coefficients are finite sets and applying the methods of topological dynamics

learning local languages and their application to dna sequence analysis this paper presents an efficient algorithm for learning in the limit a special type of regular languages called strictly locally testable languages from positive data and its application to identifying the protein spl alpha chain region in amino acid sequences first we present a linear time algorithm that given a strictly locally testable language learns its deterministic finite state automaton in the limit from only positive data this provides one with a practical and efficient method for learning a specific concept domain of sequence analysis we then describe several experimental results using the learning algorithm developed above following a theoretical observation which strongly suggests that a certain type of amino acid sequences can be expressed by a locally testable language we apply the learning algorithm to identifying the protein spl alpha chain region in amino acid sequences for hemoglobin experimental scores show an overall success rate of 95 correct identification for positive data and 96 for negative data

on the intertwined basins of attraction for planar flows in this paper we discuss the intertwined basins of attraction one of the simplest cases of obstruction to predictability some sufficient conditions to guarantee the existence of intertwined basins of attraction are given

poincare recurrence and number theory thirty years later hillel furstenberg s 1981 article in the bulletin gives an elegant introduction to the interplay between dynamics and number theory summarizing the major developments that occurred in the few years after his landmark paper 21 the field has evolved over the past thirty years with major advances on the structural analysis of dynamical systems and new results in combinatorics and number theory furstenberg s article continues to be a beautiful introduction to the subject drawing together ideas from seemingly distant fields furstenberg s article 21 gave a general correspondence between regularity properties of subsets of the integers and recurrence properties in dynamical systems now dubbed the furstenberg correspondence principle he then showed that such recurrence properties always hold proving what is now referred to as the multiple recurrence theorem combined these results gave a new proof of szemeredi s theorem 45 if s z has positive upper density then s contains arbitrarily long arithmetic progressions this proof lead to an explosion of activity in ergodic theory and topological dynamics beginning with new proofs of classic results of ramsey theory and ultimately leading to significant new combinatorial and number theoretic results the full implications of these connections have yet to be understood the approach harks back to the earliest results on recurrence in the measurable setting and in the topological setting a measure preserving system is a quadruple x b t where x denotes a set b is a algebra on x is a probability measure on x b and t x x is a measurable transformation such that t 1 a a for all a b poincare recurrence states that if x b t is a measure preserving system and a b with a 0 there exists n n such that a t na 0 a topological dynamical system is a pair x t where x is a compact metric space and t x x is a continuous map one can show that any such topological space admits a borel probability measure that preserves t in particular poincare recurrence implies recurrence in the topological setting birkhoff 13 gave a direct proof of

the primes contain arbitrarily long arithmetic progressions we prove that there are arbitrarily long arithmetic progressions of primes there are three major ingredients the first is szemeredi s theorem which asserts that any subset of the integers of positive density contains progressions of arbitrary length the second which is the main new ingredient of this paper is a certain transference principle this allows us to deduce from szemeredi s theorem that any subset of a suficiently pseudorandom set or measure of positive relative density contains progressions of arbitrary length the third ingredient is a recent result of goldston and yildirim which we reproduce here using this one may place a large fraction of the primes inside a pseudorandom set of almost primes or more precisely a pseudorandom measure concentrated on almost primes with positive relative density

prey predator systems with intertwined basins of attraction in this article we show the existence of intertwined basins of attraction for a class of prey predator systems which improves the extant results by deleting some crucial and hard testing conditions

the space of composants of an indecomposable continuum abstract the family of all composants of an indecomposable continuum is studied we investigate the equivalence relation induced on an indecomposable continuum by its partition into composants we show that up to borel bireducibility such an equivalence relation can be of only two types e 0 and e 1 the e 0 type being simple and the e 1 type being complicated as a consequence of this we show that each indecomposable continuum carries a borel probability measure which assigns 0 to each composant and 0 or 1 to each borel set which is the union of a family of composants in particular it follows that there is no borel transversal for the family of all composants this solves an old problem in the theory of continua we prove that all hereditarily indecomposable continua are of the complicated type that is they fall into the e 1 type we analyze the properties of being of type e 1 or of type e 0 we show using effective descriptive set theory that the first of these properties is analytic and so the second one is coanalytic we construct examples of continua of both types in fact we produce a family of indecomposable continua and use it to prove that these properties are complete analytic and complete coanalytic respectively hence non borel so they do not admit simple topological characterizations we also use continua from this family to show that an indecomposable continuum may be of type e 1 only because of the behavior of composants on a small subset of the continuum this in particular shows that certain natural approaches to solving kuratowski s problem on generic ergodicity of the composant equivalence relation will not work we finish with some open problems

the mobius function is strongly orthogonal to nilsequences we show that the mobius function n is strongly asymptotically or thogonal to any polynomial nilsequence f g n n2n here g is a sim ply connected nilpotent lie group with a discrete and cocompact subgroup so g is a nilmanifold g z g is a polynomial sequence and f g r is a lipschitz function more precisely we show that j 1 n p n n 1 n f g n j f g a log a n for all a 0 in particu lar this implies the mobius and nilsequence conjecture mn s from our earlier paper for every positive integer s this is one of two major in gredients in our programme to establish a large number of cases of the generalised hardy littlewood conjecture which predicts how often a collec tion 1 t z d z of linear forms all take prime values the proof is a relatively quick application of the results in our recent companion paper we give some applications of our main theorem we show for exam ple that the mobius function is uncorrelated with any bracket polynomial such as n p 3bn p 2c we also obtain a result about the distribution of nilse quences a n x n2n as n ranges only over the primes

a nilpotent roth theorem let t and s be invertible measure preserving transformations of a probability measure space x we prove that if the group generated by t and s is nilpotent then lim n ras infty frac 1 n sum n 1 n u t n x v s n x exists in l 2 norm for any u v l x we also show that for a with a 0 one has lim n ras infty frac 1 n sum n 1 n mu a cap t n a cap s n a 0 by the way of contrast we bring examples showing that if measure preserving transformations t s generate a solvable group then i the above limits do not have to exist ii the double recurrence property fails that is for some a a 0 one may have a t n a s n a 0 for all n finally we show that when t and s generate a nilpotent group of class c lim n ras infty frac 1 n sum n 1 n u t n x v s n x int ud mu int vd mu in l 2 x for all u v l x if and only if t s is ergodic on x x and the group generated by t 1 s t 2 s 2 t c s c acts ergodically on x

basins of wada abstract we describe situations in which there are several regions more than two with the wada property namely that each point that is on the boundary of one region is on the boundary of all we argue that such situations arise even in studies of the forced damped pendulum where it is possible to have three attractor regions coexisting and the three basins of attraction have the wada property

characterizations of recursively enumerable languages by means of insertion grammars abstract an insertion grammar is based on pure rules of the form uv uxv the string x is inserted in the context u v a strict subfamily of the context sensitive family is obtained incomparable with the family of linear languages we prove here that each recursively enumerable language can be written as the weak coding of the image by an inverse morphism of a language generated by an insertion grammar with the maximal length of strings u v as above equal to seven this result is rather surprising in view of some closure properties established earlier in the literature some consequences of this result are also stated when also erasing rules of the form uxv uv are present the string x is erased from the context u v then a much easier representation of recursively enumerable languages is obtained as the intersection with v of a language generated by an insertion grammar with erased strings having the maximal length of strings u v as above equal to two

global topological properties of the hopf bifurcation we study the homotopical and homological properties of the attractors evolving from a generalized hopf bifurcation we consider the lorenz equations for parameter values near the hopf bifurcation and study a natural morse decomposition of the global attractor calculating the cech homotopy type of the lorenz attractor the shape indexes of the morse sets and the morse equation of the decomposition

nilsequences and a structure theorem for topological dynamical systems abstract we characterize inverse limits of nilsystems in topological dynamics via a structure theorem for topological dynamical systems that is an analog of the structure theorem for measure preserving systems we provide two applications of the structure the first is to nilsequences which have played an important role in recent developments in ergodic theory and additive combinatorics we give a characterization that detects if a given sequence is a nilsequence by only testing properties locally meaning on finite intervals the second application is the construction of the maximal nilfactor of any order in a distal minimal topological dynamical system we show that this factor can be defined via a certain generalization of the regionally proximal relation that is used to produce the maximal equicontinuous factor and corresponds to the case of order 1

the green tao theorem on arithmetic progressions in the primes an ergodic point of view a long standing and almost folkloric conjecture is that the primes contain arbitrarily long arithmetic progressions until recently the only progress on this conjecture was due to van der corput who showed in 1939 that there are infinitely many triples of primes in arithmetic progression in an amazing fusion of methods from analytic number theory and ergodic theory ben green and terence tao showed that for any positive integer k there exist infinitely many arithmetic progressions of length k consisting only of prime numbers this is an introduction to some of the ideas in the proof concentrating on the connections to ergodic theory

on the weight of universal insertion grammars we study the computational power of pure insertion grammars we show that pure insertion grammars of weight 3 can characterize all recursively enumerable languages this is achieved by either applying an inverse morphism and a weak coding or a left right quotient with a regular language we also study an application in dna computing and improve some known results concerning the power of insertion deletion dna systems

multiple recurrence and convergence for sequences related to the prime numbers for any measure preserving system x mathcal x mu t and a in mathcal x with mu a 0 we show that there exist infinitely many primes p such that mu bigl a cap t p 1 a cap t 2 p 1 a bigr 0 furthermore we show the existence of the limit in l 2 mu of the associated double average over the primes a key ingredient is a recent result of green and tao on the von mangoldt function a combinatorial consequence is that every subset of the integers with positive upper density contains an arithmetic progression of length three and common difference of the form p 1 for some prime p

dissonant points and the region of influence of non saddle sets abstract the aim of this paper is to study dynamical and topological properties of a flow in the region of influence of an isolated non saddle set we see in particular that some topological conditions are sufficient to guarantee that these sets are attractors or repellers we study in detail the existence of dissonant points of the flow which play a key role in the description of the region of influence of a non saddle set these points are responsible for much of the dynamical and topological complexity of the system we also study non saddle sets from the point of view of the conley index theory and consider among other things the case of flows on manifolds with trivial first cohomology group for flows on these manifolds dynamical robustness is equivalent to topological robustness we carry out a particular study of 2 dimensional flows and give a topological condition which detects the existence of dissonant points for flows on surfaces we also prove that isolated invariant continua of planar flows with global region of influence are necessarily attractors or repellers

morphic characterizations of languages in chomsky hierarchy with insertion and locality this paper concerns new characterizations of regular context free and recursively enumerable languages using insertion systems with lower complexity this is achieved by using both strictly locally testable languages and morphisms the representation is in a similar way to the chomsky schutzenberger representation of context free languages specifically each recursively enumerable language l can be represented in the form l h l r where is an insertion system of weight 3 3 r is a strictly 2 testable language and h is a projection a similar representation can be obtained for context free languages using insertion systems of weight 2 0 and strictly 2 testable languages as well as for regular languages using insertion systems of weight 1 0 and strictly 2 testable languages

orbits of the pseudocircle the following theorem is proved theorem the pseudocircle has uncountably many orbits under the action of its homeomorphism group each orbit is the union of uncountably many composants a pseudocircle is a circularly chainable hereditarily indecomposable separating plane continuum r h bing 1 constructed a pseudocircle in 1951 and asked two questions are any two pseudocircles homeomorphic is a pseudocircle homogeneous in 1968 l fearnley 6 proved that the answer to the first question is yes and fearnley 5 and j t rogers jr 14 independently proved that the answer to the second question is no since the advent of the effros theorem several elegant proofs of the nonhomogeneity of the pseudocircle have appeared 7 10 and 13 in 1968 after seeing the proof that the pseudocircle is not homogeneous f b jones asked the second author how many orbits the pseudocircle had under the action of its homeomorphism group but the matter was not pursued further recently however the question has arisen again and wayne lewis 3 has asked if the pseudocircle has uncountably many orbits the purpose of this paper is to prove the following theorem which gives an affirmative answer to this question theorem the pseudocircle has uncountably many orbits under the action of its homeomorphism group each of these orbits is the union of uncountably many composants an interesting sidelight of the proof is the construction of two homeomorphic sets one being an open set of the pseudocircle and the other being an open set of the pseudoarc during the proof we construct an uncountable abelian subgroup of the homeomorphism group of the pseudocircle the homeomorphism constructed by handel 8 might be an element of this group each orbit of the pseudocircle is a borel set 4 lewis has announced that no orbit can be a g6 in the course of our argument we prove this but we do not determine further restrictions on the type of the borel set for the orbit question what classes of borel sets occur as the orbits of the pseudocircle in particular are each two orbits of the same class received by the editors may 1 1984 and in revised form august 13 1985 1980 mathematics subject classification primary 54f20 54f50

a complete classification of homogeneous plane continua we show that the only compact and connected subsets i e continua x of the plane mathbb r 2 which contain more than one point and are homogeneous in the sense that the group of homeomorphisms of x acts transitively on x are up to homeomorphism the circle mathbb s 1 the pseudo arc and the circle of pseudo arcs these latter two spaces are fractal like objects which do not contain any arcs it follows that any compact and homogeneous space in the plane has the form x z where x is either a point or one of the three homogeneous continua above and z is either a finite set or the cantor set the main technical result in this paper is a new characterization of the pseudo arc following lelek we say that a continuum x has span zero provided for every continuum c and every pair of maps f g colon c to x such that f c subset g c there exists c 0 in c so that f c 0 g c 0 we show that a continuum is homeomorphic to the pseudo arc if and only if it is hereditarily indecomposable i e every subcontinuum is indecomposable and has span zero

on the conjecture of wood and projective homogeneity abstract in 2005 kawamura and rambla independently constructed a metric counterexample to wood s conjecture from 1982 we exhibit a new nonmetric counterexample of a space l such that c 0 l c is almost transitive and show that it is distinct from a nonmetric space whose existence follows from the work of greim and rajagopalan in 1997 up to our knowledge this is only the third known counterexample to wood s conjecture we also show that contrary to what was expected if a one point compactification of a space x is r h bing s pseudo circle then c 0 x c is not almost transitive for a generic choice of points finally we point out close relation of these results on wood s conjecture to a work of irwin and solecki on projective fraisse limits and projective homogeneity of the pseudo arc and addressing their conjecture we show that the pseudo circle is not approximately projectively homogeneous

on the creation of wada basins in interval maps through fixed point tangent bifurcation basin boundaries play an important role in the study of dynamics of nonlinear models in a variety of disciplines such as biology chemistry economics engineering and physics one of the goals of nonlinear dynamics is to determine the global structure of the system such as boundaries of basins a basin having the strange property that every point which is on the boundary of that basin is on the boundary of at least three different basins is called a wada basin and its boundary is called a wada basin boundary here we consider maps on the interval we present a sufficient and necessary condition guaranteeing that three wada basins are emerging from a tangent bifurcation for certain one dimensional maps having negative schwarzian derivative two fixed point attractors on one side of the tangent bifurcation and three fixed point attractors on the other side of the tangent bifurcation all the conditions involved are numerically verifiable

an inverse theorem for the gowers u s 1 n norm we prove the inverse conjecture for the gowers u s 1 n norm for all s 1 this is new for s 4 more precisely we establish that if f n 1 1 is a function with f u s 1 n d then there is a bounded complexity s step nilsequence f g n g that correlates with f where the bounds on the complexity and correlation depend only on s and d from previous results this conjecture implies the hardy littlewood prime tuples conjecture for any linear system of finite complexity

morphic characterizations with insertion systems controlled by a context of length one among the variety of representation theorems for context free languages the chomsky schutzenberger theorem is unique in that it consists of dyck languages regular languages and simple operations in the present paper we obtain some characterizations of and representation theorems for languages in the chomsky hierarchy by using insertion systems strictly locally testable languages and morphisms in the framework of the chomsky schutzenberger theorem for instance each context free language l can be represented in the form l h l c r where c is an insertion system of weight 1 1 r is a strictly 2 testable language and h is a morphism in contrast if r is instead a strictly 1 testable language the class of languages generated is a proper subclass of context free languages and incomparable with the class of regular languages

polynomial extensions of van der waerden s and szemeredi s theorems an extension of the classical van der waerden and szemeredi the orems is proved for commuting operators whose exponents are polynomials as a consequence for example one obtains the following result let s zl be a set of positive upper banach density let p1 n pk n be polynomials with rational coefficients taking integer values on the integers and satisfying pi 0 0 i 1 k then for any v1 vk zl there exist an integer n and a vector u zl such that u pi n vi s for each i k department of mathematics ohio state university columbus ohio 43210 e mail address vitaly math ohio state edu department of mathematics technion haifa 23000 israel e mail address sashal techunix technion ac il current address department of mathematics stanford university stanford california 94305 e mail address leibman math stanford edu license or copyright restrictions may apply to redistribution see http www ams org journal terms of use