functionals of brownian motion localization and metric graphs we review several results related to the problem of a quantum particle in a random environment in an introductory part we recall how several functionals of brownian motion arise in the study of electronic transport in weakly disordered metals weak localization two aspects of the physics of the one dimensional strong localization are reviewed some properties of the scattering by a random potential time delay distribution and a study of the spectrum of a random potential on a bounded domain the extreme value statistics of the eigenvalues then we mention several results concerning the diffusion on graphs and more generally the spectral properties of the schrodinger operator on graphs the interest of spectral determinants as generating functions characterizing the diffusion on graphs is illustrated finally we consider a two dimensional model of a charged particle coupled to the random magnetic field due to magnetic vortices we recall the connection between spectral properties of this model and winding functionals of planar brownian motion

spectral zeta functions of fractals and the complex dynamics of polynomials we obtain formulas for the spectral zeta function of the laplacian on symmetric finitely ramified fractals such as the sierpinski gasket and a fractal laplacian on the interval these formulas contain a new type of zeta function associated with a polynomial rational functions also can appear in this context it is proved that this zeta function has a meromorphic continuation to a half plane with poles contained in an arithmetic progression it is shown as an example that the riemann zeta function is the zeta function of a quadratic polynomial which is associated with the laplacian on an interval the spectral zeta function of the sierpinski gasket is a product of the zeta function of a polynomial and a geometric part the poles of the former are canceled by the zeros of the latter a similar product structure was discovered by m l lapidus for self similar fractal strings

locally loop algebras and locally affine lie algebras abstract in this study we investigate a new class of lie algebras i e tame locally extended affine lie algebras of nullity 1 which are an infinite rank analog of affine lie algebras this type of algebra is called a locally affine lie algebra a certain ideal of a locally affine lie algebra called a core is a universal extension of a local version of a loop algebra which is called a locally loop algebra we classify locally loop algebras and locally affine lie algebras

quantum graphs ii some spectral properties of quantum and combinatorial graphs the paper deals with some spectral properties of mostly infinite quantum and combinatorial graphs quantum graphs have been intensively studied lately due to their numerous applications to mesoscopic physics nanotechnology optics and other areas a schnol type theorem is proven that allows one to detect that a point belongs to the spectrum when a generalized eigenfunction with an sub exponential growth integral estimate is available a theorem on spectral gap opening for decorated quantum graphs is established its analogue is known for the combinatorial case it is also shown that if a periodic combinatorial or quantum graph has a point spectrum it is generated by compactly supported eigenfunctions scars

the zeta function of the laplacian on certain fractals we prove that the zeta function of the laplacian a on self similar fractals with spectral decimation admits a meromorphic continuation to the whole complex plane we characterise the poles compute their residues and give expressions for some special values of the zeta function furthermore we discuss the presence of oscillations in the eigenvalue counting function thereby answering a question posed by j kigami and m lapidus for this class of fractals

the normalized laplacian spectrum of subdivisions of a graph determining and analyzing the spectra of graphs is an important and exciting research topic in mathematics science and theoretical computer science the eigenvalues of the normalized laplacian of a graph provide information on its structural properties and also on some relevant dynamical aspects in particular those related to random walks in this paper we give the spectra of the normalized laplacian of iterated subdivisions of simple connected graphs as an example of application of these results we find the exact values of their multiplicative degree kirchhoff index kemeny s constant and number of spanning trees

p adic dynamics the quadratic map overp adic numbers is studied in some detail we prove that near almost all indifferent fixed points it is topologically conjugate to a quasiperiodic linear map we also establish the existence of chaotic behavior and describe it using symbolic dynamics

mesures et equidistribution sur les espaces de berkovich the proof by ullmo and zhang of bogomolov s conjecture about points of small height in abelian varieties made a crucial use of an equidistribution property for small points in the associated complex abelian variety we study the analogous equidistribution property at p adic places our results can be conveniently stated within the framework of the analytic spaces defined by berkovich the first one is valid in any dimension but is restricted to algebraic metrics the second one is valid for curves but allows for more general metrics in particular to the normalized heights with respect to dynamical systems

equidistribution for torsion points of a drinfeld module we prove an equidistribution result for torsion points of drinfeld modules of generic characteristic we also show that similar equidistribution statements would provide proofs for the manin mumford and the bogomolov conjectures for drinfeld modules

universal coverings of steinberg lie algebras of small characteristic abstract it is well known that the second homology group h 2 st n r of the steinberg lie algebra st n r is trivial when n 5 in this paper we will work out h 2 st n r explicitly for n 3 4 which are not necessarily trivial consequently we obtained h 2 sl n r for n 3 4

positive line bundles on arithmetic varieties p 5 u integers is defined to count the volume of the lattice i l n of integral sections in the space j7 lon of real sections with supremum norm we want to prove that the leading term of this hilbert function as n f oo is given in terms of the height of x in as 1 this is known as a theorem of gillet and soule gs2 if x has a regular generic fiber beside this known result our proof uses hironaka s theorem on resolutions of singularities and minkowski s theorem on successive minima by hironaka s theorem we may construct

singularly continuous spectrum of a self similar laplacian on the half line we investigate the spectrum of the self similar laplacian which generates the so called pq random walk on the integer half line using the method of spectral decimation we prove that the spectral type of the laplacian is singularly continuous whenever p 12 this serves as a toy model for generating singularly continuous spectrum which can be generalized to more complicated settings we hope it will provide more insight into fibonacci type and other weakly self similar models

an ahlfors islands theorem for non archimedean meromorphic functions we present a p adic and non archimedean version of ahlfors five islands theorem for meromorphic functions extending an earlier theorem of the author for holomorphic functions in the non archimedean setting the theorem requires only four islands with explicit constants we present examples to show that the constants are sharp and that other hypotheses of the theorem cannot be removed

non archimedean holomorphic maps and the ahlfors islands theorem we present a p adic and non archimedean version of some classical complex holomorphic function theory our main result is an analogue of the five islands theorem from ahlfors theory of covering surfaces for non archimedean holomorphic maps our theorem requires only two islands with explicit and nearly sharp constants as opposed to the three islands without explicit constants in the complex holomorphic theory we also present non archimedean analogues of other results from the complex theory including theorems of koebe bloch and landau with sharp constants 0 introduction in the late nineteenth and early twentieth centuries there was a broad effort devoted to understanding the mapping properties of complex holomorphic and meromorphic functions in the mid 1930s much of that study culminated in ahlfors theory of covering surfaces the theory is sometimes considered to be a blend of nevanlinna theory and the riemann hurwitz formula or a nevanlinna theory for domains rather than for points perhaps the most elegant result of the theory is the so called five islands theorem which appeared in various preliminary forms 2 3 a few years before ahlfors main exposition of covering surfaces in 4 the statement is

phase transitions on fractals iii infinitely ramified lattices for pt ii see ibid vol 17 p 435 1984 in the first two papers of this series the authors considered self similar fractal lattices with a finite order of ramification r in the present paper they study physical models defined on a family of fractals with r infinity in order to characterise the geometry of these systems they need the connectivity q and the lacunarity l in addition to the fractal dimensionality d it is found that discrete symmetry spin models on these lattices undergo a phase transition at tc 0 an approximate renormalisation group scheme is constructed and used to find the dependence of tc and the critical exponents on the geometrical factors they also solve the problem of resistor networks on these fractals and discuss its consequences concerning spin models with continuous symmetry

spatial log periodic oscillations of first passage observables in fractals for transport processes in geometrically restricted domains the mean first passage time mfpt admits a general scaling dependence on space parameters for diffusion anomalous diffusion and diffusion in disordered or fractal media for transport in self similar fractal structures we obtain an expression for the source target distance dependence of the mfpt that exhibits both the leading power law behavior depending on the hausdorff and spectral dimension of the fractal as well as small log periodic oscillations that are a clear and definitive signal of the underlying fractal structure we also present refined numerical results for the sierpinski gasket that confirm this oscillatory behavior

the integrated density of states for the difference laplacian on the modified koch graph we consider the integrated density of statesn of the difference laplacian on the modified koch graph we show thatn increases only with jumps and a set of jump points ofn is the set of eigenvalues of with the infinite multiplicity we establish also that 0 c 1 leqslant mathop lim limits lambda to 0 frac n lambda lambda d s 2 overline mathop lim limits lambda to 0 frac n lambda lambda d s 2 leqslant c 2 infty whered s 2log5 log 40 3 is the spectral dimension of mkg

on the chaotic behavior of a generalized logistic p adic dynamical system abstract in the paper we describe basin of attraction p adic dynamical system g x a x 2 x 1 moreover we also describe the siegel discs of the system since the structure of the orbits of the system is related to the geometry of the p adic siegel discs

preperiodic points and unlikely intersections in this article we combine complex analytic and arithmetic tools to study the preperiodic points of one dimensional complex dynamical systems we show that for any fixed a b c and any integer d 2 the set of c c for which both a and b are preperiodic for z c is infinite if and only if a b this provides an affirmative answer to a question of zannier which itself arose from questions of masser concerning simultaneous torsion sections on families of elliptic curves using similar techniques we prove that if rational functions c z have infinitely many preperiodic points in common then all of their preperiodic points coincide and in particular the maps must have the same julia set this generalizes a theorem of mimar who established the same result assuming that and are defined over q the main arithmetic ingredient in the proofs is an adelic equidistribution theorem for preperiodic points over number fields and function fields with non archimedean berkovich spaces playing an essential role

fekete configuration quantitative equidistribution and wandering critical orbits in non archimedean dynamics let f be a rational function of degree d 1 on the projective line over a possibly non archimedean algebraically closed field a well known process initiated by brolin considers the pullbacks of points under iterates of f and produces an important equilibrium measure we define the asymptotic fekete property of pullbacks of points which means that they mirror the equilibrium measure appropriately as application we obtain an error estimate of equidistribution of pullbacks of points for c1 test functions in terms of the proximity of wandering critical orbits to the initial points and show that the order is o sqrt kd k upto a specific exceptional set of capacity 0 of initial points which is contained in the set of superattracting periodic points and the omega limit set of wandering critical points from the julia set or the presingular domains of f as an application in arithmetic dynamics together with a dynamical diophantine approximation these estimates recover favre and rivera letelier s quantitative equidistribution in a purely local manner

non archimedean nevanlinna theory in several variables and the non archimedean nevanlinna inverse problem cartan s method is used to prove a several variable non archimedean nevanlinna second main theorem for hyperplanes in projective space the corresponding defect relation is derived but unlike in the complex case we show that there can only be finitely many non zero non archimedean defects we then address the non archimedean nevanlinna inverse problem by showing that given a set of defects satisfying our conditions and a corresponding set of hyperplanes in projective space there exists a non archimedean analytic function with the given defects at the specified hyperplanes and with no other defects 1 history and introduction nevanlinna theory broadly speaking studies to what extent something like the fundamental theorem of algebra holds for meromorphic functions unlike polynomials transcendental meromorphic functions in general have infinitely many zeros however they have only finitely many zeros inside a disc of radius r therefore in order to study the values of a meromorphic function nevanlinna theory associates to each meromorphic function f three functions of r the distance from the origin for their precise definitions see nevanlinna ne 2 the characteristic or height function tf r measures the growth of f and should be thought of as the analogue of the degree of a polynomial the counting function nf a r counts the number of times as a logarithmic average f takes on the value a in the disc of radius r finally the mean proximity function mf a r measures how often on average f stays close to the value a on the circle of radius r nevanlinna proved two main theorems about these functions the so called first main theorem states that tf r mf a r nf a r o 1 where the bounded term o 1 depends on f and a but not on r this should be thought of as a substitute for the fundamental theorem of algebra in the following sense the first main theorem says that mf a r nf a r is essentially independent of the value a and this is analogous to the fact that a polynomial takes on every finite value the same number of times counting multiplicity the first received by the editors october 14 1995 and in revised form june 17 1996 1991 mathematics subject classification primary 11j99 11s80 30d35 32h30 32p05

asymptotics of the transition probabilities of the simple random walk on self similar graphs it is shown explicitly how self similar graphs can be obtained as blow up constructions of finite cell graphs c this yields a larger family of graphs than the graphs obtained by discretising continuous self similar fractals for a class of symmetrically self similar graphs we study the simple random walk on a cell graph c starting at a vertex v of the boundary of c it is proved that the expected number of returns to v before hitting another vertex in the boundary coincides with the resistance scaling factor using techniques from complex rational iteration and singularity analysis for green functions we compute the asymptotic behaviour of the n step transition probabilities of the simple random walk on the whole graph the results of grabner and woess for the sierpinski graph are generalised to the class of symmetrically self similar graphs and at the same time the error term of the asymptotic expression is improved finally we present a criterion for the occurrence of oscillating phenomena of the n step transition probabilities

distribution of rational maps with a preperiodic critical point let f lambda be a family of rational maps of a fixed degree with a marked critical point c lambda under a natural assumption we first prove that the hypersurfaces of parameters for which c lambda is periodic converge as a sequence of positive closed 1 1 currents to the bifurcation current attached to c and defined by demarco we then turn our attention to the parameter space of polynomials of a fixed degree d by intersecting the d 1 currents attached to each critical point of a polynomial bassaneli and berteloot obtained a positive measure mu rm bif of finite mass which is supported on the connectedness locus they showed that its support is included in the closure of the set of parameters admitting d 1 neutral cycles we show that the support of this measure is precisely the closure of the set of strictly critically finite polynomials i e of misiurewicz points

energy and laplacian on hanoi type fractal quantum graphs this article studies potential theory and spectral analysis on compact metric spaces which we refer to as fractal quantum graphs these spaces can be represented as a possibly infinite union of one dimensional intervals and a totally disconnected possibly uncountable compact set which roughly speaking represents the set of junction points classical quantum graphs and fractal spaces such as the hanoi attractor are included among them we begin with proving the existence of a resistance form on the hanoi attractor and go on to establish heat kernel estimates and upper and lower bounds on the eigenvalue counting function of laplacians corresponding to weakly self similar measures on the hanoi attractor these estimates and bounds rely heavily on the relation between the length and volume scaling factors of the fractal we then state and prove a necessary and sufficient condition for the existence of a resistance form on a general fractal quantum graph finally we extend our spectral results to a large class of weakly self similar fractal quantum graphs

universal central extensions of direct limits of lie superalgebras abstract we show that the universal central extension of a direct limit of perfect lie superalgebras l i is isomorphic to the direct limit of the universal central extensions of l i as an application we describe the universal central extensions of some infinite rank lie superalgebras

dirac and magnetic schrodinger operators on fractals abstract in this paper we define local dirac operators and magnetic schrodinger hamiltonians on fractals and prove their essential self adjointness to do so we use the concept of 1 forms and derivations associated with dirichlet forms as introduced by cipriani and sauvageot and further studied by the authors jointly with rockner ionescu and rogers for simplicity our definitions and results are formulated for the sierpinski gasket with its standard self similar energy form we point out how they may be generalized to other spaces such as the classical sierpinski carpet

the hot spots conjecture on the level 3 sierpinski gasket abstract in this paper using spectral decimation we prove that the hot spots conjecture holds on the level 3 sierpinski gasket i e every eigenfunction of the second smallest eigenvalue of the neumann laplacian introduced by j kigami attains its maximum and minimum on the boundary

harmonization and homogenization on fractals this paper suggests a direct approach to define the laplacian the spectral dimension of nested fractals and the pre sierpinski carpet conductivity we find a geometric construction of the harmonic functions on the gasket and therefore can describe effectively the dense set of functions having finite energy the paper is mostly aimed at the homogenization on the pre sierpinski gasket whose horizontal and nonhorizontal bonds have different conductivities a andb respectively we prove the convergence of the rescaled energies on the pre sierpinski gasket to a b e where e is the standard energy on the gasket with uniform conductivities we also find an explicit expression for the effective conductivity a b and deduce that its set of singularities turns out to be the julia set of a certain rational function a special section is devoted to the problem of the pre sierpinski carpet conductivity asymptotic behavior for this problem a new proof of barlow bass inequalities with sharper constants is given

equidistribution and integral points for drinfeld modules we prove that the local height of a point on a drinfeld module can be computed by averaging the logarithm of the distance to that point over the torsion points of the module this gives rise to a drinfeld module analog of a weak version of siegel s integral points theorem over number fields and to an analog of a theorem of schinzel s regarding the order of a point modulo certain primes

counting spanning trees on fractal graphs and their asymptotic complexity using the method of spectral decimation and a modified version of kirchhoff s matrix tree theorem a closed form solution to the number of spanning trees on approximating graphs to a fully symmetri

attracting cycles in p adic dynamics and height bounds for postcritically finite maps a rational function of degree at least 2 with coefficients in an algebraically closed field is postcritically finite pcf if and only if all of its critical points have finite forward orbit under iteration we show that the collection of pcf rational functions is a set of bounded height in the moduli space of rational functions over the complex numbers once the well understood family known as flexible lattes maps is excluded as a consequence there are only finitely many conjugacy classes of non lattes pcf rational maps of a given degree defined over any given number field the key ingredient of the proof is a nonarchimedean version of fatou s classical result that every attracting cycle of a rational function over c attracts a critical point

harmonic analysis for resistance forms abstract in this paper we define the green functions for a resistance form by using effective resistance and harmonic functions then the green functions and harmonic functions are shown to be uniformly lipschitz continuous with respect to the resistance metric making use of this fact we construct the green operator and the measure valued laplacian the domain of the laplacian is shown to be a subset of uniformly lipschitz continuous functions while the domain of the resistance form in general consists of uniformly 1 2 holder continuous functions

geometry of p adic siegel discs abstract we survey recent advances in the study of regular motions over p adic fields show its varied connections with dynamics and number theory and illustrate its significance to an important class of discrete dynamical systems we also show that mappings supporting quasi periodic motions can be naturally interpreted as flows with p adic time

preperiodic points of polynomials over global fields given a global field k and a polynomial defined over k of degree at least two morton and silverman conjectured in 1994 that the number of k rational preperiodic points of is bounded in terms of only the degree of k and the degree of in 1997 for quadratic polynomials over k q call and goldstine proved a bound which was exponential in s the number of primes of bad reduction of by careful analysis of the filled julia sets at each prime we present an improved bound on the order of s log s our bound applies to polynomials of any degree at least two over any global field k let k be a field and let k z be a rational function let denote the n iterate of under composition that is is the identity function and for n 1 n 1 we will study the dynamics on the projective line p k in particular we say a point x is preperiodic under if there are integers n m 0 such that x x the point y x satisfies n m y y and is said to be periodic of period n m note that x p k is preperiodic if and only if its orbit x n 0 is finite for example let k q and z z 29 16 then 5 4 1 4 7 4 forms a periodic cycle of period 3 and 5 4 1 4 7 4 and 3 4 each land on this cycle after one or two iterations in addition the point is of course fixed these nine q rational points are all preperiodic under meanwhile it is not difficult to see that no other point in p q is preperiodic by showing that the denominator of a rational preperiodic point must be 4 and that the absolute value must be less than 2 in general for any global field k any dimension n 1 and any morphism p p over k of degree at least two northcott proved in 1950 that the number of k rational preperiodic points of is finite 25 more precisely he showed that the preperiodic points form a set of bounded arithmetic height years later by analogy with the theorems of mazur 19 and merel 20 on k rational torsion of elliptic curves morton and silverman proposed the following conjecture 23 uniform boundedness conjecture morton and silverman 1994 given integers d n 1 and d 2 there is a constant d n d with the following property let k be a number field with k q d and let p p be a morphism of degree d defined over k then has at most preperiodic points in p k the analogy between preperiodic points and torsion comes from the fact that the torsion points of an elliptic curve e are precisely the preperiodic points of the multiplication bytwo map 2 e e in fact taking x coordinates the map 2 induces a rational function known to dynamicists as a lattes map p p of degree 4 whose preperiodic points date june 21 2005 revised december 21 2005 2000 mathematics subject classification primary 11g99 secondary 11d45 37f10

on lifting commutative dynamical systems we construct a lifting of commuting power series from a residue field of positive characteristic to a local ring in a manner similar to the formal group cohomologies of lubin tate this lifting is then used to prove that a nontorsion zp automorphism and a multiplication by p endomorphism uniquely define a formal group over fields of a positive characteristic this suggests that a characteristic 0 conjecture by lubin could be approached with residue field considerations

bifurcation measures and quadratic rational maps we study critical orbits and bifurcations in the curves per1 within the moduli space m2 of quadratic rational maps f p 1 p 1 for each 2 c the curve per1 is dened by the condition that the map has a xed point of multiplier we prove that the curve per1 contains innitely many postcritically n ite maps if and only if 0 addressing a special case of bd2 conjecture 1 4 we also show that the two critical points of f dene distinct bifurcation measures along per 1

central extensions of lie algebras graded by finite root systems lie algebras graded by finite irreducible reduced root systems have been classified up to central extensions by berman and moody benkart and zelmanov and neher in this paper we determine the central extensions of these lie algebras and hence describe them completely up to isomorphism

the hot spots conjecture for the sierpinski gasket abstract there are many works on the hot spots conjecture for domains in euclidean space since the conjecture was posed by j rauch in 1974 in this paper using spectral decimation we prove that the conjecture holds on the sierpinski gasket i e every eigenfunction of the second smallest eigenvalue of the neumann laplacian introduced by j kigami attains its maximum and minimum on the boundary

locally extended affine lie algebras abstract we propose a new simplified definition of extended affine lie algebras ealas for short and also discuss a general version of extended affine lie algebras called locally extended affine lie algebras local ealas for short we prove a conjecture by v kac for local ealas it turns out that the root system of a local eala becomes a locally finite version of an extended affine root system several examples of new ealas and local ealas are introduced and finally we classify local ealas of nullity 0 and show the connection to locally finite split simple lie algebras

on the spectrum of the normalized laplacian of iterated triangulations of graphs the eigenvalues of the normalized laplacian of a graph provide information on its topological and structural characteristics and also on some relevant dynamical aspects specifically in relation to random walks in this paper we determine the spectra of the normalized laplacian of iterated triangulations of a generic simple connected graph as an application we also find closed forms for their multiplicative degree kirchhoff index kemeny s constant and number of spanning trees

sur la compacite des ensembles de julia des polynomes p adiques let r be a rational function of degree 2 with complex coefficients then the julia set of r is a closed subset of 1 and therefore compact if one replace by the field p completion of an algebraic closure of the field p of the p adic numbers then one can define also a julia set for a rational function with p adic coefficients but as p is not locally compact the julia set may or may not be compact in this paper we study the compactness of the julia set of p adic polynomials

cycles of quadratic polynomials and rational points on a genus 2 curve it has been conjectured that for n sufficiently large there are no quadratic polynomials in q z with rational periodic points of period n morton proved there were none with n 4 by showing that the genus 2 algebraic curve that classifies periodic points of period 4 is birational to x 1 16 whose rational points had been previously computed we prove there are none with n 5 here the relevant curve has genus 14 but it has a genus 2 quotient whose rational points we compute by performing a 2 descent on its jacobian and applying a refinement of the method of chabauty and coleman we hope that our computation will serve as a model for others who need to compute rational points on hyperelliptic curves we also describe the three possible galois stable 5 cycles and show that there exist galois stable n cycles for infinitely many n furthermore we answer a question of morton by showing that the genus 14 curve and its quotient are not modular finally we mention some partial results for n 6

a finiteness theorem for canonical heights attached to rational maps over function fields let k be a function field let k t be a rational map of degree d 2 defined over k and suppose that is not isotrivial in this paper we show that a point p p k has canonical height zero if and only if p is preperiodic for this answers affirmatively a question of szpiro and tucker and generalizes a recent result of benedetto from polynomials to rational functions we actually prove the following stronger result which is a variant of the northcott finiteness principle there exists e 0 such that the set of points p p k with canonical height at most e is finite our proof is essentially analytic making use of potential theory on berkovich spaces to study the dynamical green s functions g v x y attached to at each place v of k for example we show that every conjugate of has bad reduction at v if and only if g v x x 0 for all x p1berk v where p 1 berk v denotes the berkovich projective line over the completion of kv in an appendix we use a similar method to give a new proof of the mordell weil theorem for elliptic curves over k

spectral dimension and bohr s formula for schrodinger operators on unbounded fractal spaces we establish an asymptotic formula for the eigenvalue counting function of the schrodinger operator for some unbounded potentials v on several types of unbounded fractal spaces we give sufficient conditions for bohr s formula to hold on metric measure spaces which admit a cellular decomposition and then verify these conditions for fractafolds and fractal fields based on nested fractals in particular we partially answer a question of fan khandker and strichartz regarding the spectral asymptotics of the harmonic oscillator potential on the infinite blow up of a sierpinski gasket

what is the alternative to the alexander orbach relation the alexander orbach ao relation d w 2d f d s connecting the fractal dimension of a random walk s rw trajectory d w or the exponent of anomalous diffusion 2 d w on a fractal structure with the fractal and spectral dimension of the structure itself plays a key role in discussion of dynamical properties of complex systems including living cells and single biomolecules this relation however does not hold universally and breaks down for some structures like diffusion limited aggregates and eden trees we show that the alternative to the ao relation is the explicit dependence of the coefficient of the anomalous diffusion on the system s size i e the absence of its thermodynamical limit the prerequisite for its breakdown is the dependence of the local structure of possible steps of the rw on the system s size the discussion is illustrated by the examples of diffusion on a koch curve ao conform and on a cantor dust violating ao relation

the resolvent kernel for pcf self similar fractals for the laplaciandefined on a p c f self similar fractal we give an explicit formula for the resolvent kernel of the laplacian with dirichlet boundary conditions and also with neumann boundary conditions that is we construct a symmetric function g which solves i 1 f x r g x y f y d y the method is similar to kigami s construction of the green kernel in kig01 3 5 and is expressed as a sum of scaled and translated copies of a certain function which may be considered as a fundamental solution of the resolvent equation examples of the explicit resolvent kernel formula are given for the unit interval standard sierpinski gasket an d the level 3 sierpinski gasket s g3

canonical heights transfinite diameters and polynomial dynamics let z be a polynomial of degree at least 2 with coefficients in a number field k iterating gives rise to a dynamical system and a corresponding canonical height function as defined by call and silverman we prove a simple product formula relating the transfinite diameters of the filled julia sets of over various completions of k and we apply this formula to give a generalization of bilu s equidistribution theorem for sequences of points whose canonical heights tend to zero

explicit formulas for heat kernels on diamond fractals this paper provides explicit pointwise formulas for the heat kernel on compact metric measure spaces that belong to a mathbb n times mathbb n parameter family of fractals which are regarded as projective limits of metric measure graphs and do not satisfy the volume doubling property the formulas are applied to obtain uniform continuity estimates of the heat kernel and to derive an expression of the fundamental solution of the free schrodinger equation the results also open up the possibility to approach infinite dimensional spaces based on this model

on the asymptotics of the eigenvalue counting function for random recursive sierpinski gaskets we consider natural laplace operators on random recursive affine nested fractals based on the sierpinski gasket and prove an analogue of weyl s classical result on their eigenvalue asymptotics the eigenvalue counting function n is shown to be of order ds 2 as where we can explicitly compute the spectral dimension d s moreover the limit n ds 2 will typically exist and can be expressed as a deterministic constant multiplied by a random variable this random variable is a power of the limiting random variable in a suitable general branching process and has an interpretation as the volume of the fractal

diffusion on the scaling limit of the critical percolation cluster in the diamond hierarchical lattice we construct critical percolation clusters on the diamond hierarchical lattice and show that the scaling limit is a graph directed random recursive fractal a dirichlet form can be constructed on the limit set and we consider the properties of the associated laplace operator and diffusion process in particular we contrast and compare the behaviour of the high frequency asymptotics of the spectrum and the short time behaviour of the on diagonal heat kernel for the percolation clusters and for the underlying lattice in this setting a number of features of the lattice are inherited by the critical cluster

locally semisimple and maximal subalgebras of the finitary lie algebras gl sl so and sp abstract we describe all locally semisimple subalgebras and all maximal subalgebras of the finitary lie algebras gl sl so and sp for simple finite dimensional lie algebras these classes of subalgebras have been described in the classical works of a malcev and e dynkin

the size of quadratic p adic linearization disks we find the exact radius of linearization disks at indifferent fixed points of quadratic maps in cp we also show that the radius is invariant under power series perturbations localizing all periodic orbits of these quadratic like maps we then show that periodic points are not the only obstruction for linearization in so doing we provide the first known examples in the dynamics of polynomials over cp where the boundary of the linearization disk does not contain any periodic point

pure point spectrum of the laplacians on fractal graphs abstract we establish the pure point spectrum of the laplacians on two point self similar fractal graphs all eigenvalues have infinite multiplicity and a countable system of orthonormal eigenfunctions with compact support is complete in the corresponding hilbert space

spectral analysis on infinite sierpinski gaskets we study the spectral properties of the laplacian on infinite sierpin ski gaskets we prove that the laplacian with the neumann boundary condition has pure point spectrum moreover the set of eigenfunctions with compact support is complete the same is true if the infinite sierpin ski gasket has no boundary but is false for the laplacian with the dirichlet boundary condition in all these cases we describe the spectrum of the laplacian and all the eigenfunctions with compact support to obtain these results first we prove certain new formulas for eigenprojectors of the laplacian on finite sierpin ski pre gaskets then we prove that the spectrum of the discrete laplacian on a sierpin ski lattice is pure point and the eigenfunctions are localized 1998 academic press

hyperfunctions and spectral zeta functions of laplacians on self similar fractals we investigate the spectral zeta function of fractal differential operators such as the laplacian on the unbounded i e infinite sierpinski gasket and a self similar sturm liouville operator associated with a fractal self similar measure on the half line in the latter case sabot discovered the relation between the spectrum of this operator and the iteration of a rational map of several complex variables called the renormalization map we obtain a factorization of the spectral zeta function of such an operator expressed in terms of the dirac delta hyperfunction a geometric zeta function and the zeta function associated with the dynamics of the corresponding renormalization map viewed either as a polynomial function on in the first case or in the second case as a polynomial on the complex projective plane our first main result extends to the case of the fractal laplacian on the unbounded sierpinski gasket a factorization formula obtained by the second author for the spectral zeta function of a fractal string and later extended by teplyaev to the bounded i e finite sierpinski gasket and some other decimable fractals furthermore our second main result generalizes these factorization formulas to the renormalization maps of several complex variables associated with fractal sturm liouville operators moreover as a corollary in the very special case when the underlying self similar measure is the lebesgue measure on 0 1 we obtain a representation of the riemann zeta function in terms of the dynamics of a certain polynomial in thereby extending to several variables an analogous result by teplyaev

weyl s problem for the spectral distribution of laplacians on p c f self similar fractals we establish an analogue of weyl s classical theorem for the asymptotics of eigenvalues of laplacians on a finitely ramified i e p c f self similar fractalk such as for example the sierpinski gasket we consider both dirichlet and neumann boundary conditions as well as laplacians associated with bernoulli type multifractal measures onk from a physical point of view we study the density of states for diffusions or for wave propagation in fractal media more precisely let x be the number of eigenvalues less thanx then we show that x is of the order ofxds 2 asx where the spectral exponent ds is computed in terms of the geometric as well as analytic structures ofk further we give an effective condition that guarantees the existence of the limit ofx ds 2 x asx this condition is in some sense generic in addition we define in terms of the above spectral exponents and calculate explicitly the spectral dimension ofk

on minimal decomposition of p adic homographic dynamical systems abstract a homographic map in the field of p adic numbers q p is studied as a dynamical system on p 1 q p the projective line over q p if such a system admits one or two fixed points in q p then it is conjugate to an affine dynamics whose dynamical structure has been investigated by fan and fares in this paper we shall mainly solve the remaining case that the system admits no fixed point we shall prove that this system can be decomposed into a finite number of minimal subsystems which are topologically conjugate to each other all the minimal subsystems are exhibited and the unique invariant measure for each minimal subsystem is determined

lie algebras graded by 3 graded root systems and jordan pairs covered by grids let r be an irreducible root system a lie algebra l is called graded by r if l is graded with grading group the root lattice of r such that the only nonzero homogeneous subspaces of l have degree 0 or a root in r the grading is induced by the adjoint action of a split cartan subalgebra of a finite dimensional simple lie subalgebra of l with root system r and l is generated by the homogeneous subspaces of nonzero degree this class of lie algebras was introduced and studied by s berman and r moody in invent math 108 1992 where in particular a classification up to central equivalence is given in the simply laced case the doubly laced cases have recently been classified by g benkart and e zelmanov let r be a 3 graded root system i e r is not of type e 8 f 4 or g 2 in this paper lie algebras graded by r are described in a unified way without case by case considerations namely it is shown that a lie algebra l is 3 graded if and only if l is a central extension of the tits kantor koecher algebra of a jordan pair covered by a grid whose associated 3 graded root system is isomorphic to r this result is then used to classify lie algebras graded by r we give the classification of jordan pairs covered by a grid and describe their tits kantor koecher algebras one of the advantages of this approach is that it works over rings containing 1 2 and 1 3 and also for infinite root systems another application is the description of slodowy s intersection matrix algebras arising from multiply affinized cartan matrices

phase transitions on fractals i quasi linear lattices magnetic spin models and resistor networks are studied on certain self similar fractal lattices which are described as quasi linear because they share a significant property of the line finite portions can be isolated from the rest by removal of two points sites in all cases there is no long range order at finite temperature the transition at zero temperature has a discontinuity in the magnetisation and the associated magnetic exponent is equal to the fractal dimensionality d when the lattice reduces to a non branching curve the thermal exponent v 1 y is equal to d when the lattice is a branching curve y is related respectively to the dimensionality of the single channel segments of the curve for the ising model or to the exponent describing the resistivity for models with continuous spin symmetry

nonarchimedean green functions and dynamics on projective space let varphi mathbb p n k to mathbb p n k be a morphism of degree d 2 defined over a field k that is algebraically closed field and complete with respect to a nonarchimedean absolute value we prove that a modified green function hat g varphi associated to varphi is holder continuous on mathbb p n k and that the fatou set mathcal f varphi of varphi is equal to the set of points at which hat g phi is locally constant further hat g varphi vanishes precisely on the set of points p such that varphi has good reduction at every point in the forward orbit mathcal o varphi p of p we also prove that the iterates of varphi are locally uniformly lipschitz on mathcal f varphi

vibration modes of 3n gaskets and other fractals we rigorously study eigenvalues and eigenfunctions vibration modes on the class of self similar symmetric finitely ramified fractals which include the sierpinski gasket and other 3n gaskets we consider the classical laplacian on fractals which generalizes the usual one dimensional second derivative is the generator of the self similar diffusion process and has possible applications as the quantum hamiltonian we develop a theoretical matrix analysis including analysis of singularities which allows us to compute eigenvalues eigenfunctions and their multiplicities exactly we support our theoretical analysis by symbolic and numerical computations our analysis in particular allows the computation of the spectral zeta function on fractals and the limiting distribution of eigenvalues i e integrated density of states we consider such examples as the level 3 sierpinski gasket a fractal 3 tree and the diamond fractal

siegel s theorem for drinfeld modules we prove a siegel type statement for finitely generated phi submodules of mathbb g a under the action of a drinfeld module phi this provides a positive answer to a question we asked in a previous paper we also prove an analog for drinfeld modules of a theorem of silverman for nonconstant rational maps of mathbb p 1 over a number field