self organized partially synchronous dynamics in populations of nonlinearly coupled oscillators we analyze a minimal model of a population of identical oscillators with a nonlinear coupling a generalization of the popular kuramoto model in addition to well known for the kuramoto model regimes of full synchrony full asynchrony and integrable neutral quasiperiodic states ensembles of nonlinearly coupled oscillators demonstrate two novel nontrivial types of partially synchronized dynamics self organized bunch states and self organized quasiperiodic dynamics the analysis based on the watanabe strogatz ansatz allows us to describe the self organized bunch states in any finite ensemble as a set of equilibria and the self organized quasiperiodicity as a two frequency quasiperiodic regime an analytic solution in the thermodynamic limit of infinitely many oscillators is also discussed

powers property and simplec algebras would be desirable to understand the ideal structure of b in terms of the data less ambitiously we are interested here in criteria for b to be simple this is a popular problem it is for example discussed for f abelian at the end of 13 or for a outer in the introduction of 5 the present work concentrates on cases where f is a so called powers group we recall the definition after proposition 2 below but where ct is arbitrary recall from 8 that the following are powers groups 1 non abelian free groups and more generally free products which are neither trivial nor z 2z z 2z 2 non amenable subgroups of psl 2 c 3 sl 3 z and more generally subgroups of sl 3 c containing sl 3 z for the two next theorems ct is an action of a powers group f on a c algebra a with 1 and

null controllability with constraints on the state for the reaction diffusion system this paper is addressed to study the null controllability with constraints on the state for the reaction diffusion system we transform the problem into an equivalent null controllability problem with constraint on the control using a carleman inequality adapted to the constraint we prove the null controllability with constraints on the state for the reaction diffusion system holds

mapping tori of free group automorphisms are coherent the mapping torus of an endomorphism of a group g is the hnnextensiong g with bonding maps the identity and we show that a mapping torus of an injective free group endomorphism has the property that its flnitely generated subgroups are flnitely presented and moreover these subgroups are of flnite type

c c simplicity and the amenable radical a countable group is c simple if its reduced c algebra is simple it is well known that c simplicity implies that the amenable radical of the group must be trivial we show that the converse does not hold by constructing explicit counter examples we additionally prove that every countable group embeds into a countable group with trivial amenable radical and that is not c simple

time phase bispectral analysis bispectral analysis a technique based on high order statistics is extended to encompass time dependence for the case of coupled nonlinear oscillators it is applicable to univariate as well as to multivariate data obtained respectively from one or more of the oscillators it is demonstrated for a generic model of interacting systems whose basic units are the poincare oscillators their frequency and phase relationships are explored for different coupling strengths both with and without gaussian noise the distinctions between additive linear or quadratic and parametric frequency modulated interactions in the presence of noise are illustrated

pseudo anosov maps and invariant train tracks in the disc a finite algorithm let f be an isotopy class of orientation preserving homeomorphisms of the punctured disc we construct a finite algorithm which enables one to decide whether the class f contains a pseudo anosov element in this case the algorithm defines an invariant train track and the dilatation factor which gives the minimal topological entropy of the isotropy class

stable actions of groups on real trees this paper further develops rips s work on real trees we study a class of actions called stable which includes actions with trivial arc stabilizers and small actions of hyperbolic groups

parageometric outer automorphisms of free groups we study those fully irreducible outer automorphisms o of a finite rank free group f r which are parageometric meaning that the attracting fixed point of o in the boundary of outer space is a geometric r tree with respect to the action of f r but o itself is not a geometric outer automorphism in that it is not represented by a homeomorphism of a surface our main result shows that the expansion factor of o is strictly larger than the expansion factor of o 1 as corollaries proved independently by guirardel the inverse of a parageometric outer automorphism is neither geometric nor parageometric and a fully irreducible outer automorphism o is geometric if and only if its attracting and repelling fixed points in the boundary of outer space are geometric r trees

correct initial boundary value problems for dispersive equations this paper deals with correctness of initial boundary value problems for general dispersive equations of finite odd orders for the kawahara and kdv equations we prove existence uniqueness and stability of strong global solutions in a bounded domain for different signs of a coefficient of the highest derivative as well as their asymptotics when the coefficient of the higher order derivative in the kawahara equation approaches zero

new proofs of some results of nielsen given a compact oriented surface m with negative euler characteristic and a homeomorphism z m2 m2 what is the best representative cp m2 m2 in the isotopy class of s thurston s answered this question by defining pseudo anosov homeomorphisms a generalization of anosov diffeomorphisms on t2 which may be characterized as follows the homeomorphism rp is pseudo anosov iff there is a pair of transverse measured geodesic laminations i and au which intersect each closed nonperipheral geodesic in h4 which are topologically preserved by cp and which have their respective transverse measures multiplied under the action of cp by l n and d for some l 1 thurston showed that cp m m2 can be pieced together from homeomorphisms of subsurfaces which are either periodic or pseudoanosov more precisely thurston proved theorem 0 1 the preprint s provides an introduction to this theorem describing many of the useful properties of pseudo anosov diffeomorphisms the proof outlined in s is contained in 111

boundary value problems for the stationary kawahara equation abstract this paper deals with boundary value problems in a bounded domain for the stationary one dimensional kawahara equation for reasonable boundary conditions we prove existence and uniqueness of a strong solution and its asymptotics when the coefficient of a higher order derivative approaches zero formulations of well posed and ill posed problems for odd order equations are discussed

onset of synchronization in large networks of coupled oscillators we study the transition from incoherence to coherence in large networks of coupled phase oscillators we present various approximations that describe the behavior of an appropriately defined order parameter past the transition and generalize recent results for the critical coupling strength we find that under appropriate conditions the coupling strength at which the transition occurs is determined by the largest eigenvalue of the adjacency matrix we show how with an additional assumption a mean field approximation recently proposed is recovered from our results we test our theory with numerical simulations and find that it describes the transition when our assumptions are satisfied we find that our theory describes the transition well in situations in which the mean field approximation fails we study the finite size effects caused by nodes with small degree and find that they cause the critical coupling strength to increase

wavelet bispectral analysis for the study of interactions among oscillators whose basic frequencies are significantly time variable bispectral analysis recently introduced as a technique for revealing time phase relationships is extended to make use of wavelets rather than fourier analysis it is thus able to encompass instantaneous phase time dependence for the case of two or more coupled nonlinear oscillators the method is demonstrated and evaluated by use of test signals from a pair of coupled poincare oscillators it promises to be useful in a wide range of scientific contexts for studies of interacting oscillators whose basic frequencies are significantly time variable

rank 1 phenomena for mapping class groups we prove that every element of the mapping class group g has linear growth confirming a conjecture of n ivanov and that g is not boundedly generated we also provide restrictions on linear representations of g and its finite index subgroups

direction of coupling from phases of interacting oscillators an information theoretic approach a directionality index based on conditional mutual information is proposed for application to the instantaneous phases of weakly coupled oscillators its abilities to distinguish unidirectional from bidirectional coupling as well as to reveal and quantify asymmetry in bidirectional coupling are demonstrated using numerical examples of quasiperiodic chaotic and noisy oscillators as well as real human cardiorespiratory data

detection of weak directional coupling phase dynamics approach versus state space approach we compare two conceptually different approaches to the detection of weak directional couplings between two oscillatory systems from bivariate time series the first approach is based on the analysis of the systems phase dynamics whereas the other one tests for interdependencies in the reconstructed state spaces of the systems we analyze the sensitivity of both techniques to weak couplings in numerical experiments by considering couplings between almost identical as well as between significantly different nonlinear systems we study different degrees of phase diffusion test the robustness of the two techniques against observational noise and investigate the influence of the time series length our results show that none of the two approaches is generally superior to the other and we conclude that it is probably the combination of both techniques that would allow the most comprehensive and reliable characterization of coupled systems

directionality of coupling from bivariate time series how to avoid false causalities and missed connections we discuss some problems encountered in inference of directionality of coupling or in the case of two interacting systems in inference of causality from bivariate time series we identify factors and influences that can lead to either decreased test sensitivity or false detections and propose ways to cope with them in order to perform tests with high sensitivity and a low rate of false positive results

the equivalence of some residual properties of word hyperbolic groups abstract we show that every word hyperbolic group is residually finite if and only if every word hyperbolic group has a finite quotient

phase synchronization of chaotic attractors in the presence of two competing periodic signals we discuss the situation where two periodic signals compete to phase synchronize a chaotic attractor depending on the relative position of the periods with respect to the synchronization tongue for a single frequency signal we distinguish several different cases we find that depending on parameters it is possible that one or the other signal will entrain exclusively or that they will entrain alternately at their average frequency or not at all

phase relationships between two or more interacting processes from one dimensional time series i basic theory a general approach is developed for the detection of phase relationships between two or more different oscillatory processes interacting within a single system using one dimensional time series only it is based on the introduction of angles and radii of return times maps and on studying the dynamics of the angles an explicit unique relationship is derived between angles and the conventional phase difference introduced earlier for bivariate data it is valid under conditions of weak forcing this correspondence is confirmed numerically for a nonstationary process in a forced van der pol system a model describing the angles behavior for a dynamical system under weak quasiperiodic forcing with an arbitrary number of independent frequencies is derived to establish from experimental data whether or not two or more interacting processes are synchronized is an old and important problem in the absence of noise the existence of synchronization between two processes in interacting limit cycle oscillators was originally taken to imply that their basic frequencies of oscillation are related as integer numbers i e are rationally connected 1 2 and that their instantaneous phases are permanently locked this definition suggests that the detection of whether or not synchronization exists can be established by computation of the ratio of basic frequencies in the fourier spectrum of the signal from one of the sub systems involved even in this simplest case however the finite observation time and the discreteness of the digitiza tion steps used in practice will make it appear that all fre quencies are rationally connected thereby complicating the reliable estimation of their ratio furthermore the noise that is invariably present in all real macroscopic systems means that only effective synchronization 3 can normally take place meaning that the phases can remain locked only dur ing finite time intervals and that the basic frequencies may no longer be rationally connected 4 serious difficulties may also arise due to the nonstationarity of experimental data in view of these problems modern techniques for estab lishing the presence or absence of synchronization are based on the assumption that the behavior of each subsystem can be considered separately and that their individual time series can be compared by a variety of techniques e g by compu tation of the phase difference between them this approach has been justified theoretically 3 6 and is widely used to detect synchronization not only in periodic noisy but also in

the automorphism group of a free group is not linear a linear group has ascending chain condition on centralizers malcev s p 511 finitely generated linear groups are residually finite malcev s p 453 a solvable linear group is nilpotent by abelian byfinite tits 5 pp 14551461 a linear group either contains a free group of rank two or is solvable by locally finite lubotzky 3 has characterized finitely generated linear groups over a field of characteristic zero by purely group theoretic conditions these conditions appear difficult to check although they have been used to show linearity for certain groups however we do not use his theorem the main result of this paper theorem 5 is that for n 2 3 the automorphism group of a free group of rank n is not a linear group the proof uses the representation theory of algebraic groups to show that a kind of diophantine equation between the irreducible representations of

the korteweg de vries kawahara equation in a bounded domain and some numerical results abstract we are concerned with the initial boundary problem associated to the korteweg de vries kawahara perturbed by a dispersive term which appears in several fluids dynamics problems we obtain local smoothing effects that are uniform with respect to the size of the interval we also propose a simple finite different scheme for the problem and prove its unconditional stability finally we give some numerical examples

la dynamique des pseudogroupes de rotations we study dynamical systems on the circle generated by a finite number of partially defined rotations we construct new examples with all orbits dense this leads to non simplicial free actions of free groups on trees we study the generic dynamics for these pseudogroups and their 1 parameter families we show that in suitable 2 parameter families the set of pseudogroups having a dense orbit is a sierpinski curve we generalize results on interval exchange transformations obtained by boshernitzan veech rips

stability of the homology of the mapping class groups of orientable surfaces the mapping class group of f fgs r is f rgs wo a where a is the topological group of orientation preserving diffeomorphisms of f which are the identity on df and fix the s punctures when r 0 and 2g s 3 an important feature of f is that it acts properly discontinuously on the teichmiller space us the quotient is the moduli space f of isometry classes of complete hyperbolic metrics of finite area on f since us is homeomorphic to euclidean space and f is virtually torsion free we have

automorphisms of automorphism groups of free groups abstract if n 3 and f n is free of rank n then out aut f n out out f n 1

detecting the harmonics of oscillations with time variable frequencies a method is introduced for the spectral analysis of complex noisy signals containing several frequency components it enables components that are independent to be distinguished from the harmonics of nonsinusoidal oscillatory processes of lower frequency the method is based on mutual information and surrogate testing combined with the wavelet transform and it is applicable to relatively short time series containing frequencies that are time variable where the fundamental frequency and harmonics of a process can be identified the characteristic shape of the corresponding oscillation can be determined enabling adaptive filtering to remove other components and nonoscillatory noise from the signal thus the total bandwidth of the signal can be correctly partitioned and the power associated with each component then can be quantified more accurately the method is first demonstrated on numerical examples it is then used to identify the higher harmonics of oscillations in human skin blood flow both spontaneous and associated with periodic iontophoresis of a vasodilatory agent the method should be equally relevant to all situations where signals of comparable complexity are encountered including applications in astrophysics engineering and electrical circuits as well as in other areas of physiology and biology

inferring the directionality of coupling with conditional mutual information uncovering the directionality of coupling is a significant step in understanding drive response relationships in complex systems in this paper we discuss a nonparametric method for detecting the directionality of coupling based on the estimation of information theoretic functionals we consider several different methods for estimating conditional mutual information the behavior of each estimator with respect to its free parameter is shown using a linear model where an analytical estimate of conditional mutual information is available numerical experiments in detecting coupling directionality are performed using chaotic oscillators where the influence of the phase extraction method and relative frequency ratio is investigated

automorphisms of free groups have finitely generated fixed point sets an automorphism of a finitely generated free group f extends to a homeomorphism of the end completion e of f the set of fixed points of this homeomorphism is finitely generated in a certain sense in particular this implies that the subgroup of elements fixed by the automorphism is finitely generated in the usual sense the emphasis on e instead of f in this paper is analogous to thurston s study of surface groups where measured laminations are studied instead of simple closed curves the techniques in this paper arose from a study of the behaviour of an automorphism under iteration that is the dynamics of the automorphism see 11 i thank bill thurston for explaining some of his ideas to me for analysing automorphisms of free groups in particular the result on bounded cancellation below is due to thurston and matt grayson peter scott asked whether the fixed subgroup is always finitely generated in 3 and gersten first answered this question in 2 using quite different methods

method to control the coupling function using multilinear feedback methods to control the dynamics of coupled oscillators have been developed owing to various medical and technological demands in this study we develop a method to control coupled oscillators in which the coupling function expressed in a phase model is regulated by the multilinear feedback the present method has wide applicability because we do not need to measure an individual output from each oscillator but only measure the sum of the outputs from all the oscillators moreover it allows us to easily control the coupling function up to higher harmonics the validity of the present method is confirmed through a simulation

some metric properties of automorphisms of groups abstract study of the dynamics of automorphisms of a group is usually focused on their growth and or finite orbits including fixed points in this paper we introduce properties of a different kind using somewhat informal language we call them metric properties two principal characteristics of this kind are called here the curl and the flux there seems to be very little correlation between these and the growth of an automorphism which means they are likely to be an essentially new tool for studying automorphisms we also observe that our definitions of the curl and flux are sufficiently general to be applied to mappings of arbitrary metric spaces

instantaneous frequency and amplitude identification using wavelets application to glass structure this paper describes a method for extracting rapidly varying superimposed amplitude modulated and frequency modulated signal components the method is based upon the continuous wavelet transform cwt and uses a new wavelet that is a modification to the well known morlet wavelet to allow analysis at high resolution in order to interpret the cwt of a signal correctly an approximate analytic expression for the cwt of an oscillatory signal is examined via a stationary phase approximation this analysis is specialized for the new wavelet and the results are used to construct expressions for the amplitude and frequency modulations of the components in a signal from the transform of the signal the method is tested on a representative variable frequency signal as an example before being applied to a function of interest in our subject area a structural correlation function of a disordered material which immediately reveals previously undetected features

automorphisms of free groups have asymptotically periodic dynamics we show that every automorphism alpha of a free group f k of finite rank k has it asymptotically periodic dynamics on f k and its boundary partial f k there exists a positive power alpha q such that every element of the compactum f k cup partial f k converges to a fixed point under iteration of alpha q

an algorithm to identify automorphisms which arise from self induced interval exchange transformations we give an algorithm to determine if the dynamical system generated by a positive automorphism of the free group can also be generated by a self induced interval exchange transformation the algorithm effectively yields the interval exchange transformation in case of success

stable representatives for symmetric automorphisms of groups and the general form of the scott conjecture let g be a group acting on a tree x such that all edge stabilizers are finite we extend bestvina handel s theory of train tracks for automorphisms of free groups to automorphisms of g which permute vertex stabilizers using this extension we show that there is an upper bound depending only on g for the complexity of the graph of groups decomposition of the fixed subgroups of such automorphisms of g

the symmetries of outer space for n 3 the natural map out fn aut kn from the outer automorphism group of the free group of rank n to the group of simplicial automorphisms of the spine of outer space is an isomorphism

cyclic splittings of finitely presented groups and the canonical jsj decomposition the classification of stable actions of finitely presented groups on trees has found a number of applications perhaps one of the most striking of these applications is the theory of canonical jsj decompositions of hyperbolic groups developed by sela in his seminal paper se

detecting couplings between interacting oscillators with time varying basic frequencies instantaneous wavelet bispectrum and information theoretic approach in the natural world the properties of interacting oscillatory systems are not constant but evolve or fluctuating continuously in time thus the basic frequencies of the interacting oscillators are time varying which makes the system analysis complex for studying their interactions we propose a complementary approach combining wavelet bispectral analysis and information theory we show how these methods uncover the interacting properties and reveal the nature strength and direction of coupling wavelet bispectral analysis is generalized as a technique for detecting instantaneous phase time dependence for the case of two or more coupled nonlinear oscillators whereas the information theory approach can uncover the directionality of coupling and extract driver response relationships in complex systems we generate bivariate time series numerically to mimic typical situations that occur in real measured data apply both methods to the same time series and discuss the results the approach is applicable quite generally to any system of coupled nonlinear oscillators

cubulating hyperbolic free by cyclic groups the irreducible case let v be a nite graph and let v v be an irreducible train track map whose mapping torus has word hyperbolic fundamental group g then g acts freely and cocompactly on a cat 0 cube complex hence if f is a nite rank free group and f f an irreducible monomorphism so that g f is word hyperbolic then g acts freely and cocompactly on a cat 0 cube complex this holds in particular if is an irreducible automorphism with g f o z word hyperbolic

mapping class groups and outer automorphism groups of free groups are c simple abstract we prove that the reduced c algebras of centerless mapping class groups and outer automorphism groups of free groups are simple as are the irreducible pure subgroups of mapping class groups and the analogous subgroups of outer automorphism groups of free groups

the expansion factors of an outer automorphism and its inverse a fully irreducible outer automorphism o of the free group f n of rank n has an expansion factor which often differs from the expansion factor of o 1 nevertheless we prove that the ratio between the logarithms of the expansion factors of o and o 1 is bounded above by a constant depending only on the rank n we also prove a more general theorem applying to an arbitrary outer automorphism of f n and its inverse and their two spectrums of expansion factors

acylindrical accessibility for groups we define the notion of acylindrical graph of groups of a group we bound the combinatorics of these graphs of groups for f g freely indecomposable groups our arguments imply the finiteness of acylindrical surfaces in closed 3 manifolds ha finiteness of isomorphism classes of small splittings of torsion free freely indecomposable hyperbolic groups as well as finiteness results for small splittings of f g kleinian and semisimple discrete groups acting on non positively curved simply connected manifolds in order to get our accessibility for f g groups we generalize parts of rips analysis of stable actions of f p groups on real trees to f g groups the concepts we present play an essential role in constructing the canonical jsj decomposition se1 ri se2 in obtaining the hopf property for hyperbolic groups se2 and in our study of sets of solutions to equations in a free group se3

the kawahara equation on bounded intervals and on a half line abstract we consider an initial boundary value problem for the kawahara equation posed on bounded intervals and on a half line we prove the existence and uniqueness of global regular and weak solutions on bounded intervals 0 l as well as a priori estimates which do not depend on l and prove that regular and weak solutions on bounded intervals converge to regular and weak solutions on the positive half line as l

topology of finite graphs this paper derives from a course in group theory which i gave at berkeley in 1982 i wanted to prove the standard theorems on free groups and discovered that after a few preliminaries the notion of locally injective map or immersion of graphs was very useful this enables one to see in an effective easy algorithmic way just what happens with finitely generated free groups one can understand in this way 1 howson s theorem that if a and b are finitely generated subgroups of a free group then a b is finitely generated and 2 m hall s theorem that finitely generated subgroups of free groups are closed in the profinite topology during this course s m gersten came up with a simple proof of h neumann s inequality on the ranks in howson s theorem one of the ideas in gersten s proof was to use core graphs graphs with no trees hanging on subsequently i found that some consequences of a paper of greenberg s could be proved using core graphs and covering translations of immersions the most striking such result is that if a and b are finitely generated subgroups of a free group and if a c b is of finite index in both a and b then a b is of finite index in a v b the subgroup generated by a ub

existence of perturbed solitary wave solutions to a model equation for water waves abstract we prove the existence of travelling wave solutions to a fifth order partial differential equation which is a formal asymptotic approximation for water waves with surface tension these travelling waves are arbitrarily small perturbations of solitary waves but are not solitary waves themselves because they approach small amplitude oscillations for large values of the independent variable this result suggests that for bond numbers less than one third there are branches of travelling wave solutions to the water wave equations which are perturbations of supercritical elevation solitary waves and which bifurcate from froude number one and bond number one third

detecting automorphic orbits in free groups abstract we present an effective algorithm for detecting automorphic orbits in free groups as well as a number of algorithmic improvements of train tracks for free group automorphisms

strichartz estimates for dispersive equations and solvability of the kawahara equation abstract we first establish a series of strichartz estimates for a general class of linear dispersive equations by applying the theory of oscillatory integrals established by kenig ponce and vega next we use such estimates to study solvability of the cauchy problem of the kawahara equation t u a u x u x 3 u x 5 u 0 in the class c r h s r local existence is proved for s 1 4 and global existence is proved for s 2

on the cubical geometry of higman s group we investigate the cocompact action of higman s group on a cat 0 square complex associated to its standard presentation we show that this action is in a sense intrinsic which allows for the use of geometric techniques to study the endomorphisms of the group and show striking similarities with mapping class groups of hyperbolic surfaces outer automorphism groups of free groups and linear groups over the integers we compute explicitly the automorphism group and outer automorphism group of higman s group and show that the group is both hop an and co hop an we actually prove a stronger rigidity result about the endomorphisms of higman s group every non trivial morphism from the group to itself is an automorphism we also study the geometry of the action and prove a surprising result although the cat 0 square complex acted upon contains uncountably many ats the higman group does not contain subgroups isomorphic to z finally we show that this action possesses features reminiscent of negative curvature which we use to prove a re ned version of the tits alternative for higman s group higman s group was constructed in 15 as the rst example of a nitely presented in nite group without non trivial nite quotients it is given by the following presentation h ai i z 4z aiai 1a i a 2 i 1 i z 4z if some partial results about higman s group are known the structure of the group and its geometry have remained mostly elusive the group is known to split as an amalgamated product over several non abelian free subgroups and such splittings can be used to derive various properties of the group for instance the group has nite asymptotic dimension by work of bell dranishnikov 1 a property that implies the novikov conjecture and the coarse baum connes conjecture for the group 36 37 such splittings have also been used to show that the higman group admits in a sense many quotients schupp developed small cancellation tools to show that the group is sq universal 30 that is every countable group embeds in a quotient of h minasyan osin used these splittings to show that the group is 2010 mathematics subject classi cation primary 20f65 secondary 20f28

decay properties for solutions of fifth order nonlinear dispersive equations abstract we consider the initial value problem associated to a large class of fifth order nonlinear dispersive equations this class includes several models arising in the study of different physical phenomena our aim is to establish special space decay properties of solutions to these systems these properties complement previous unique continuation results and in some case show that they are optimal these decay estimates reflect the parabolic character of these dispersive models in exponential weighted spaces this principle was first obtained by t kato in solutions of the kdv equation

tits alternatives for graph products we discuss various types of tits alternative for subgroups of graph products of groups and prove that under some natural conditions a graph product of groups satisfies a given form of tits alternative if and only if each vertex group satisfies this alternative as a corollary we show that every finitely generated subgroup of a graph product of virtually solvable groups is either virtually solvable or large as another corollary we prove that every non abelian subgroup of a right angled artin group has an epimorphism onto the free group of rank 2 in the course of the paper we develop the theory of parabolic subgroups which allows to describe the structure of subgroups of graph products that contain no non abelian free subgroups we also obtain a number of results regarding the stability of some group properties under taking graph products

on the dynamics and the fixed subgroup of a free group automorphism on considere des automorphismes f n f n ou f n f a 1 a n est le groupe libre de base a 1 a 2 a n n 2 on compose un champ vectoriel sur un graphe d on introduit des aretes repulsives attractives et des points fixes a l infini et on etudie la dynamique resultante

internal controllability of the kawahara equation on a bounded domain abstract in this paper we consider the kawahara equation posed on a bounded interval with a distributed control first we establish a carleman estimate for the kawahara equation with internal observation then applying this carleman estimate we can show that the kawahara equation is null controllable furthermore we prove the null controllability with constraints on the state for the kawahara equation

the nielsen thurston classification and automorphisms of a free group i using train tracks and invariant laminations thurston has developed a whole theory to understand the dynamics and geometry of diffeomorphisms of surfaces th ca bl by introducing a combinatorial analogue of train tracks m bestvina and m handel be ha have managed to analyze irreducible automorphisms of a free group and using this analysis to bound the rank of the fixed subgroup of an automorphism by the rank of the ambient group which was known before as the scott conjecture in se1 we introduced a dynamical algebraic commutative diagram associated with a limit action of the ambient free group fn on some real tree this commutative diagram allows one to interpret dynamical invariants of the limit action in terms of algebraic properties of the automorphism in question and vice versa using this diagram we showed how to obtain the nielsen thurston classification of automorphisms of surfaces on the algebraic level and some generalized versions of the scott conjecture in this paper we make an extensive use of the dynamical algebraic commutative diagram and the classification of stable actions of finitely presented groups on real trees ri be fe1 to construct a canonical hierarchical decomposition of a free group associated with an automorphism of it which is from our point of view the analogue of the nielsen thurston classification for automorphisms of free groups it is also our belief that our hierarchical decomposition should serve as a complementary to the bestvina handel train tracks and invariant laminations in studying the dynamics and combinatorics of general rather than irreducible automorphisms of a free group