combinatorial characterization of sub hyperbolic rational maps in 1980 s thurston established a combinatorial characterization for post critically finite rational maps among post critically finite branched coverings of the two sphere to itself a completed proof was written by douady and hubbard in their paper a douady j h hubbard a proof of thurston s topological characterization of rational functions acta math 171 1993 263 297 this criterion was then extended by cui jiang and sullivan to sub hyperbolic rational maps among sub hyperbolic semi rational branched coverings of the two sphere to itself the goal of this paper is to present a new but simpler proof for the combinatorial characterization of sub hyperbolic rational maps by adapting some arguments in the proof in douady and hubbard s paper

on diffuse reflection at the boundary for the boltzmann equation and related equations the paper considers diffuse reflection at the boundary with nonconstant boundary temperature and unbounded velocities the solutions obtained are proved to conserve mass at the boundary after a preliminary study of the collisionless case the main results obtained are existence for the boltzmann equation in a diperna lions framework with the above boundary conditions in a bounded measure sense and existence together with uniqueness for the bgk equation with maxwellian diffusion on the boundary in anl framework

weak amenability of fourier algebras and local synthesis of the anti diagonal abstract we show that for a connected lie group g its fourier algebra a g is weakly amenable only if g is abelian our main new idea is to show that weak amenability of a g implies that the anti diagonal g g g 1 g g is a set of local synthesis for a g g we then show that this cannot happen if g is non abelian we conclude for a locally compact group g that a g can be weakly amenable only if it contains no closed connected non abelian lie subgroups in particular for a lie group g a g is weakly amenable if and only if its connected component of the identity g e is abelian

hyper tauberian algebras and weak amenability of figa talamanca herz algebras abstract we study certain commutative regular semisimple banach algebras which we call hyper tauberian algebras we first show that they form a subclass of weakly amenable tauberian algebras then we investigate the basic and hereditary properties of them moreover we show that if a is a hyper tauberian algebra then the linear space of bounded derivations from a into any banach a bimodule is reflexive we apply these results to the figa talamanca herz algebra a p g of a locally compact group g for p 1 we show that a p g is hyper tauberian if the principal component of g is abelian finally by considering the quantization of these results we show that for any locally compact group g a p g equipped with an appropriate operator space structure is a quantized hyper tauberian algebra this in particular implies that a p g is operator weakly amenable

the combinatorial rigidity conjecture is false for cubic polynomials we show that there exist two cubic polynomials with connected julia sets which are combinatorially equivalent but not topologically conjugate on their julia sets this disproves a conjecture by mcmullen from 1995

weak and cyclic amenability for fourier algebras of connected lie groups abstract using techniques of non abelian harmonic analysis we construct an explicit non zero cyclic derivation on the fourier algebra of the real a x b group in particular this provides the first proof that this algebra is not weakly amenable using the structure theory of lie groups we deduce that the fourier algebras of connected semisimple lie groups also support non zero cyclic derivations and are likewise not weakly amenable our results complement earlier work of johnson 1994 15 plymen 2001 18 and forrest samei and spronk 2009 9 as an additional illustration of our techniques we construct an explicit non zero cyclic derivation on the fourier algebra of the reduced heisenberg group providing the first example of a connected nilpotent group whose fourier algebra is not weakly amenable

a characterization of hyperbolic rational maps we give a topological characterization of rational maps with disconnected julia sets our results extend thurston s characterization of postcritically finite rational maps in place of iteration on teichmuller space we use quasiconformal surgery and thurston s original result

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

weak amenability for fourier algebras of 1 connected nilpotent lie groups a special case of a conjecture raised by forrest and runde 2005 10 asserts that the fourier algebra of every non abelian connected lie group fails to be weakly amenable this was already known to hold in the non abelian compact cases by earlier work of johnson 1994 13 and plymen unpublished note in recent work choi and ghandehari 2014 4 the authors verified this conjecture for the real ax b group and hence by structure theory for any semisimple lie group in this paper we verify the conjecture for all 1 connected non abelian nilpotent lie groups by reducing the problem to the case of the heisenberg group as in our previous paper an explicit non zero derivation is constructed on a dense subalgebra and then shown to be bounded using harmonic analysis en route we use the known fusion rules for schrodinger representations to give a concrete realization of the dual convolution for this group as a kind of twisted operator valued convolution we also give some partial results for solvable groups which give further evidence to support the general conjecture

renormalizations and wandering jordan curves of rational maps we realize a dynamical decomposition for a post critically finite rational map which admits a combinatorial decomposition we split the riemann sphere into two completely invariant subsets one is a subset of the julia set consisting of uncountably many jordan curve components most of them are wandering the other consists of components that are pullbacks of finitely many renormalizations together with possibly uncountably many points the quotient action on the decomposed pieces is encoded by a dendrite dynamical system we also introduce a surgery procedure to produce post critically finite rational maps with wandering jordan curves and prescribed renormalizations

on the cycles of components of disconnected julia sets for any integers d ge 3 and n ge 1 we construct a hyperbolic rational map of degree d such that it has n cycles of the connected components of its julia set except single points and jordan curves

a decomposition theorem for herman maps abstract in 1980s thurston established a topological characterization theorem for postcritically finite rational maps in this paper a decomposition theorem for a class of postcritically infinite branched covering termed herman map is developed it s shown that every herman map can be decomposed along a stable multicurve into finitely many siegel maps and thurston maps such that the combinations and rational realizations of these resulting maps essentially dominate the original one this result is motivated by a non expanding version of mcmullen s problem and thurston s theory on characterization of rational maps it enables us to prove a thurston type theorem for rational maps with herman rings

real laminations and the topological dynamics of complex polynomials abstract we characterize the laminations associated to complex polynomials with connected julia sets and without irrationally neutral cycles

on the connection between sets of operator synthesis and sets of spectral synthesis for locally compact groups we extend the results by froelich and spronk and turowska on the connection between operator synthesis and spectral synthesis for a g to second countable locally compact groups g this gives us another proof that one point subset of g is a set of spectral synthesis and that any closed subgroup is a set of local spectral synthesis furthermore we show that non triangular sets are strong operator ditkin sets and we establish a connection between operator ditkin sets and ditkin sets these results are applied to prove that any closed subgroup of g is a local ditkin set

measure solutions of the steady boltzmann equation in a slab it is shown that the steady boltzmann equation in a slab 0 a has solutionsx x such that the ingoing boundary measures 0 0 and 0 can be prescribed a priori the collision kernel is truncated such that particles with smallx component of the velocity have a reduced collision rate

initial boundary value problems for the boltzmann equation global existence of weak solutions for an open set of 3 bounded or not we consider initial boundary value problems for the boltzmann equation for general gas surface interaction laws and for hard potentials we prove a global existence result for weak solutions the proof uses the regularization of the collision operator and the renormalization method for the regularized problem by using weak compactness in l1 and averaged stability ofq f f we prove the existence of weak solutions of our problem

on the cauchy problem for boltzmann equations global existence and weak stability we study the large data cauchy problem for boltzmann equations with general collision kernels we prove that sequences of solutions which satisfy only the physically natural a priori bounds converge weakly in l to a solution from this stability result we deduce global existence of a solution to the cauchy problem our method relies upon recent compactness results for velocity averages a new formulation of the boltzmann equation which involves nonlinear normalization and an analysis of subsolutions and supersolutions it allows us to overcome the lack of strong a priori estimates and define a meaningful collision operator for general configurations

trend to equilibrium of weak solutions of the boltzmann equation in a slab with diffusive boundary conditions recently r illner and the author proved that under a physically realistic truncation on the collision kernel the boltzmann equation in the one dimensional slab 0 1 with general diffusive boundary conditions at 0 and 1 has a global weak solution in the traditional sense here it is proved that when the maxwellians associated with the boundary conditions atx 0 andx 1 are the same maxwellianmw then the solution is uniformly bounded and tends tomw fort

a global existence theorem for the initial boundary value problem for the boltzmann equation when the boundaries are not isothermal we extend the existence theorem recently proved by hamdache for the initial boundary value problem for the nonlinear boltzmann equation in a vessel with isothermal boundaries to more general situations including the case when the boundaries are not isothermal in the latter case a cut off for large speeds is introduced in the collision term of the boltzmann equation

geometrically finite and semi rational branched coverings of the two sphere in 1982 thurston gave a necessary and sufficient condition for a critically finite branched covering of the two sphere to itself to be combinatorially equivalent to a rational map we discuss extending this result to geometrically finite rational maps we give an example to show that thurston s original condition is not sufficient this example is topologically pathological near accumulation points of the postcritical set we give two conditions forbidding such pathology show that they are equivalent and in a sequel to the present paper will show that thurston s condition together with this tameness is both necessary and sufficient to characterize geometrically finite

global weak solutions of the boltzmann equation in a slab with diffusive boundary conditions it is proved that under a physically realistic truncation of the collision kernel the boltzmann equation in the one dimensional slab 0 1 with general diffusive boundary conditions at 0 and 1 has a global weak solution in the traditional sense this solution satisfies the boundary conditions almost everywhere and has at worst exponentially growing total energy

global topology of hyperbolic components cantor circle case we prove that in the moduli space md of degree d 2 rational maps any hyperbolic component in the disconnectedness locus and of cantor circle type is a finite quotient of r4d 4 n tn where n is determined by dynamics the proof uses some ideas from riemann surface theory abel s theorem dynamical system and algebraic topology

are there more than five linearly independent collision invariants for the boltzmann equation the problem of finding the summational collision invariants for the boltzmann equation is tackled with the aim of proving that the most general solution of the problem is not different from the standard one even when the equation defining a collision invariant is only satisfied almost everywhere inr3 r3 s2 the collision invariant is assumed to be in the hilbert spaceh of the functions which are square integrable with respect to a maxwellian weight

a proof of thurston s topological characterization of rational functions the criterion proved in this paper was stated by thurston in november 1982 thurston lectured on its proof on several occasions notably at the nsf summer conference in duluth 1983 where one of the authors jhh was present using the notes of various attendants at these lectures we have reconstructed a proof that we have made as precise as we could since this required a certain amount of work on our part we thought it might be of some use to present this proof to the reader we thank dennis sullivan for useful conversations and fritz yon haesseler and especially ben wittner for help with the writing and valuable suggestions after the first version was written clifford earle pointed out that better estimates than what we had were to be found in b

amenability and weak amenability of the fourier algebra let g be a locally compact group we show that its fourier algebra a g is amenable if and only if g has an abelian subgroup of finite index and that its fourier stieltjes algebra b g is amenable if and only if g has a compact abelian subgroup of finite index we then show that a g is weakly amenable if the component of the identity of g is abelian and we prove some partial results towards the converse