invariant palm and related disintegrations via skew factorization let g be a measurable group with haar measure acting properly on a space s and measurably on a space t then any finite jointly invariant measure m on s t admits a disintegration nu otimes mu into an invariant measure on s and an invariant kernel from s to t here we construct and by a general skew factorization which extends an approach by rother and zahle for homogeneous spaces s over g this leads to easy extensions of some classical propositions for invariant disintegration previously known in the homogeneous case the results are applied to the palm measures of jointly stationary pairs where is a random measure on s and is a random element in t

convex invariant means and a pathwise central limit theorem on etudie le probleme de choisir un point aleatoirement a partir de la ligne des reels on definit une notion de distribution d espace uniforme on considere trois sources supplementaires d uniformite l invariance d echelle convexe la consistance avec des methodes de moyennes et la linearite mesure de mokobodski

local entropy averages and projections of fractal measures we show that for families of measures on euclidean space which satisfy an ergodic theoretic form of self similarity under the operation of re scaling the dimension of linear images of the measure behaves in a semi continuous way we apply this to prove the following conjecture of furstenberg let m n be integers which are not powers of the same integer and let x y be closed subsets of the unit interval which are invariant respectively under times m mod 1 and times n mod 1 then for any non zero t dim x ty min 1 dim x dim y a similar result holds for invariant measures and gives a simple proof of the rudolph johnson theorem our methods also apply to many other classes of conformal fractals and measures as another application we extend and unify results of peres shmerkin and nazarov and of moreira concerning projections of products self similar measures and gibbs measures on regular cantor sets we show that under natural irreducibility assumptions on the maps in the ifs the image measure has the maximal possible dimension under any linear projection other than the coordinate projections we also present applications to bernoulli convolutions and to the images of fractal measures under differentiable maps

random recursive construction of self similar fractal measures the noncompact case the self similarity of sets measures is often defined in a constructive way in the present paper it will be shown that the random recursive construction model of falconer graf and mauldin williams for statistically self similar sets may be generalized to the noncompact case we define a sequence of random finite measures which converges almost surely to a self similar random limit measure under certain conditions on the generating lipschitz maps we determine the carrying dimension of the limit measure

sample path properties of self similar processes with stationary increments a real valued process x x t u is self similar with exponent h h ss if x a d ahx for all a 0 sample path properties of h ss processes with stationary increments are investigated the main result is that the sample paths have nowhere bounded variation if 0 1 in which case they are singular however nowhere bounded variation may occur also for h 1 examples exhibiting this combination of properties are constructed as well as many others most are obtained by subordination of random measures to point processes in ir2 that are poincar6 i e invariant in distribution for the transformations t x at b ax of r in a final section it is shown that the self similarity and stationary increment properties are preserved under composition of independent processes x1 a x2 x1 x2 t l some interesting examples are obtained this way

equidistribution from fractal measures we give a fractal geometric condition for a measure on 0 1 to be supported on points x that are normal in base n i e such that n kx k in mathbb n equidistributes modulo 1 this condition is robust under c 1 coordinate changes and it applies also when n is a pisot number rather than an integer as applications we obtain new results and strengthen old ones about the prevalence of normal numbers in fractal sets and new results on measure rigidity specifically completing host s theorem to multiplicatively independent integers and proving a rudolph johnson type theorem for certain pairs of beta transformations

convergence under q of p invariant measures on the circle abstract consider the semigroup of maps tn n n where tn x nx mod 1 suppose p and q are multiplicatively independent integers in that they are not both powers of the same integer further suppose that is a borel probability measure invariant ergodic and of positive entropy for tp we show that the sequence of measures tiq converges weak to lebesgue measure on a subsequence of values i of uniform full density

optimal control of evolutionary quasi newtonian fluids we are interested in the necessary optimality conditions for optimal control problems of systems describing the unsteady flow of incompressible shear thickening fluids to overcome the difficulties related to the differentiability of the control to state mapping consequence of the lack of regularity of the state variable we first establish some intermediate optimality conditions and then pass to the limit

a local version of the projection theorem we prove the following theorem suppose that 1 m n are integers and is a borel measure on r such that for a e x 1 the upper and lower m densities of at x are positive and finite 2 if is a tangent measure of at x then for all v g n m the orthogonal projection of the support of onto v is a convex set then is m rectifiable

statistically self similar fractals letx be a complete separable bounded metric space and a borel probability measure on the space con x n of alln tuples of contractions ofx with the topology of pointwise convergence then there exists a unique self similar probability measurep on the spacek x of all non empty compact subsets ofx here a measurep onk x is called self similar if for every borel setb k x p b int p n left k o k n 1 left bigcup limits i 0 n 1 s i k i in b right right d mu s o s n 1 if for a e s 0 sn 1 eachs i has an inverse which satisfies a lipschitz condition then there is an 0 such that forp a e k k x the hausdorff dimensionh dim k is equal to ifx d is compact and has non empty interior and if a e s 0 sn 1 consists of similarities which satisfy a certain disjointness condition w r t x then is determined by the equation int sum limits i 0 n 1 lip s i alpha d mu s o s n 1 1 where lip s i denotes the smallest lipschitz constant fors i under fairly general assumptions the dimensional hausdorff measure ofp a e k k x equals o if andx are chosen in a rather special way thenp a e k k x is the graph of a homeomorphism of 0 1 or a curve or the graph of a continuous function

tangent measure distributions of fractal measures tangent measure distributions provide a natural tool to study the local geometry of fractal sets and measures in euclidean spaces the idea is loosely speaking to attach to every point of the set a family of random measures called the alpha dimensional tangent measure distributions at the point which describe asymptotically the alpha dimensional scenery seen by an observer zooming down towards this point this tool has been used by bandt ba and graf g to study the regularity of the local geometry of self similar sets but in this paper we show that its scope goes much beyond this situation and in fact it may be used to describe a strong regularity property possessed by every measure we show that for every measure mu on a euclidean space and any dimension alpha at mu almost every point all alpha dimensional tangent measure distributions are palm measures this means that the local geometry of every dimension of general measures can be described like the local geometry of self similar sets by means of a family of statistically self similar random measures we believe that this result reveals a wealth of new and unexpected information about the structure of such general measures and we illustrate this by pointing out how it can be used to improve or generalize recently proved relations between ordinary and average densities

self similar random measures a set is called self similar if it is decomposable into parts which are similar to the whole this notion was generalized to random sets in the present paper an alternative axiomatic approach is given which makes precise the following idea using palm distribution theory a random set is statistically self similar if it is statistically scale invariant with respect to any center chosen at random from that set for these sets hausdorff dimension coincides with an intrinsic self similarity index

invariant measures and arithmetic quantum unique ergodicity we classify measures on the locally homogeneous space i sl 2 r l which are invariant and have positive entropy under the diagonal subgroup of sl 2 r and recurrent under l this classification can be used to show arithmetic quantum unique ergodicity for compact arithmetic surfaces and a similar but slightly weaker result for the finite volume case other applications are also presented in the appendix joint with d rudolph we present a maximal ergodic theorem related to a theorem of hurewicz which is used in theproof of the main result

optimal control for an adsorbate induced phase transition model this paper is concerned with the optimal control problem for an adsorbate induced phase transition model that is we show the existence of the optimal control and derive the optimality conditions by showing the differentiability of the cost functional with respect to the control

invariant transports of stationary random measures and mass stationarity we introduce and discuss balancing invariant transports of stationary random measures on a polish abelian group the first main result is an associated fundamental invariance property of palm measures derived from a generalization of neveu s exchange formula the second main result is a simple sufficient and necessary criterion for the existence of balancing invariant transports we then introduce in a non stationary setting the concept of mass stationarity with respect to a random measure formalizing the intuitive idea that the origin is a typical location in the mass the third main result of the paper is that a measure is a palm measure if and only if it is mass stationary

optimality conditions for distributed control problems with nonlinear state equation we develop an abstract approximating process for control problems with convex cost governed by evolution equations involving monotone operators applications are given concerning necessary conditions of optimality for distributed control problems with hyperbolic parabolic and delay differential state systems these conditions are expressed by means of the clarke 11 generalized gradient

an analogy of bol s result on jacobi forms and siegel modular forms in this paper the analogy of bol s result to the several variable function case is discussed one shows how to construct siegel modular forms and jacobi forms of higher degree respectively using bol s result

first and second order optimality conditions for a class of optimal control problems with quasilinear elliptic equations a class of optimal control problems for quasilinear elliptic equations is considered where the coefficients of the elliptic differential operator depend on the state function first and second order optimality conditions are discussed for an associated control constrained optimal control problem main emphasis is laid on second order sufficient optimality conditions to this aim the regularity of the solutions to the state equation and its linearization is studied in detail and the pontryagin maximum principle is derived one of the main difficulties is the nonmonotone character of the state equation

dimension growth for iterated sumsets we study dimensions of sumsets and iterated sumsets and provide natural conditions which guarantee that a set f subseteq mathbb r satisfies overline dim mathrm b f f overline dim mathrm b f or even dim mathrm h n f rightarrow 1 our results apply to for example all uniformly perfect sets which include ahlfors david regular sets our proofs rely on hochman s inverse theorem for entropy and the assouad and lower dimensions play a critical role we give several applications of our results including an erd s volkmann type theorem for semigroups and new lower bounds for the box dimensions of distance sets for sets with small dimension

multilinear operators on siegel modular forms of genus 1 and 2 classically there are many interesting connections between differential operators and the theory of elliptic modular forms and many interesting results have been explored in particular it has been known for some time how to obtain an elliptic modular form from the derivatives ofn elliptic modular forms which has already been studied in detail by r rankin in 9 and 10 when n 2 as a special case of rankin s result in 10 h cohen constructed certain covariant bilinear operators which he used to obtain modular forms with interesting fourier coefficients 4 later these covariant bilinear operators were called rankin cohen operators by d zagier who studied their algebraic relations 12 13 as well as connections with formal pseudodifferential operators 5 furthermore rankin cohen operators are shown to appear as the various terms in the convergent expansion of the composition of two symbols in a certain symbolic calculus associated with sl 2 r 11 recently the rankin cohen type bracket operator on jacobi forms and siegel modular forms has been studied using the heat operator and differential operator respectively in fact it was shown how to construct explicitly jacobi forms and siegel modular forms of genus 2 with the rankin cohen type bracket operators involving the heat operator and the determinant differential operators respectively 1

b expansions and symbolic dynamics given a real number p i it is easy to expand x e o i in a canonical way into a series of negative powers of p with integer coefficients less than or equal to p the form of the set of such p expansions which is a matter of symbolic dynamics or language theory varies according to arithmetical properties of p but there are many common features known facts which are few and open questions much more numerous are reviewed in this paper

optimal control of quasilinear parabolic equations this paper deals with optimal control problems governed by quasilinear parabolic equations in divergence form whose cost functional is of lagrangian type our aim is to prove the existence of solutions and derive some optimality conditions to attain this second objective we accomplish the sensitivity analysis of the state equation with respect to the control proving that under some assumptions this relation is g teaux differentiable finally a regularising procedure along with ekeland s variational principle allow us to treat some other problems for which this differentiability property cannot be stated

galton watson iterated function systems iterated function systems ifs are interesting parametric models for generating fractal sets and functions the general idea is to compress deform and translate a given set or function with a collection of operators and to iterate the procedure under weak assumptions ifs possess a unique fixed point which is in general fractal ifs were introduced in a deterministic context then were generalized to the random setting on abstract spaces in the early 1980 s their adaptation to random signals was carried out by hutchinson and r schendorff 9 by considering random operators this study extends their model with not only random operators but also a random underlying construction tree we show that the corresponding ifs converges under certain hypothesis to a unique fractal fixed point properties of the fixed point are also described

entropy of convolutions on the circle given ergodic p invariant measures f ig on the 1 torus w e give a sharp condition on their entropies guaranteeing that the entropy of the convolution 1 n converges to logp we also prove a variant of this result for joinings of full entropy on in conjunction with a method of host this yields the following denote aeq x qx mod 1 then for every p invariant ergodic with positive entropy 1 n p ni1 n 0 aecn converges weak to lebesgue measure as n i 1 under a certain mild combinatorial condition onfckg for instance the condition is satisfled if p 10 and ck 2 k 6 k or ck 2 2 k this extends a result of johnson and rudolph who considered the sequence ck q k when p and q are multiplicatively independent we also obtain the following corollary concerning hausdorfi dimension of sum sets for any sequence fsig of p invariant closed subsets of i f p dimh si j log dimh si j 1 then dimh s1 sn i 1

on self similar sets with overlaps and inverse theorems for entropy we study the dimension of self similar sets and measures on the line we show that if the dimension is less than the generic bound of minf1 sg where s is the similarity dimension then there are superexponentially close cylinders at all small enough scales this is a step towards the conjecture that such a dimension drop implies exact overlaps and conrms it when the generating similarities have algebraic coecients as applications we prove furstenberg s conjecture on projections of the one dimensional sierpinski gasket and achieve some progress on the bernoulli convolutions problem and more generally on problems about parametric families of self similar measures the key tool is an inverse theorem on the structure of pairs of probability measures whose mean entropy at scale 2 n has only a small amount of growth under convolution

optimal control problems governed by some semilinear parabolic equations abstract this paper is concerned with the optimal control problem for the keller segel equations that is as a generalization of ryu and yagi j math anal appl 256 2001 45 66 we derive the optimality conditions for the optimal control problem governed by a semilinear abstract equation of nonmonotone type moreover we prove the uniqueness of the optimal control under some conditions

characterization of palm measures via bijective point shifts the paper considers a stationary point process n in r d a point map picks a point of n in a measurable way it is called bijective thorisson h 2000 coupling stationarity and regeneration springer new york if it is generating by suitable shifts a bijective mapping on n mecke math nachr 65 1975 335 344 proved that the palm measure of n is point stationary in the sense that it is invariant under bijective point shifts our main result identifies this property as being characteristic for palm measures this generalizes a fundamental classical result for point processes on the line see e g theorem 11 4 in kallenberg o 2002 foundations of modern probability 2nd ed springer new york and solves a problem posed in thorisson h 2000 coupling stationarity and regeneration springer new york and ferrari p a landim c and thorisson h 2004 ann inst h poincare probab statist 40 141 152 our second result guarantees the existence of bijective point maps that have almost surely with respect to the palm measure of n no fixed points this answers another question asked by thorisson our final result shows that there is a directed graph with vertex set n that is defined in a translation invariant way and whose components are almost surely doubly infinite paths this generalizes and complements one of the main results in holroyd a e and peres y 2003 electron comm prohab 8 17 27 no additional assumptions as ergodicity nonlattice type conditions or a finite intensity are made in this paper

critical branching in a highly fluctuating random medium a model of one dimensional critical branching superprocess is constructed in a random medium fluctuating both in time and space the medium describes a moving system of point catalysts and branching occurs only in the presence of these catalysts although the medium has an infinite overall density the clumping features of the branching model can be exhibited by rescaling time space and mass by an exactly calculated scaling power which is stronger than in the constant medium case the main technique used is the asymptotic analysis of a generalized diffusion reaction equation in the space time random medium which given the medium prescribes the evolution of the laplace transition functional of the markov branching process

distributed control of systems governed by a general class of quasilinear elliptic equations abstract in this paper we are concerned with optimal control problems governed by a general class of quasilinear elliptic equations our aim is to prove existence of solution and derive optimality conditions in order to deal with this last question we begin investigating differentiability of state solution of equation with respect to the control second term of equation obtaining it only in certain cases a perturbation method is used to treat other cases as a consequence of the optimality system regularity of optimal control and state is establised

nonconvex optimal control of nonlinear monotone parabolic systems abstract an optimal control problem is studied for distributed systems governed by nonlinear parabolic pde s with state constraints the state equation is monotone in the state variable and nonlinear in the control variable the constraints and the cost functional are not necessarily convex relaxed controls are used to prove the existence of an optimal control moreover a minimum principle of relaxed optimality is established

branching random walk with exponentially decreasing steps and stochastically self similar measures we consider a branching random walk on r whose step size decreases by a fixed factor 0 1 2 the limit measure is almost surely a s absolutely continuous with respect to the lebesgue measure but for pisot 1 it is a s singular 2 for all a 5 1 2 the support of the measure is a s the closure of its interior 3 for pisot 1 the support of the measure is fractured it is a s disconnected and the components of the complement are not isolated on both sides

some local approximations of dawson watanabe superprocesses let be a dawson watanabe superprocess in rd such that t is a s locally finite for every t 0 then for d 2 and fixed t 0 the singular ran dom measure t can be a s approximated by suitably normalized restrictions of lebesgue measure to the neighborhoods of suppff when d 3 the local distributions of near a hitting point can be approximated in total vari ation by those of a stationary and self similar pseudo random measure f by contrast the corresponding distributions for d 2 are locally invariant fur ther results include improvements of some classical extinction criteria and some limiting properties of hitting probabilities our main proofs are based on a detailed analysis of the historical structure of f

self similar random measures are locally scale invariant in an earlier paper patzschke and u zahle 11 have proved the existence of a fractional tangent measure at the typical point of a self similar random measure under rather special technical assumptions in the present paper we remove the most restrictive one here we suppose the open set condition for the similarities a constant positive lower bound for the random contraction ratios and vanishing on the boundary of the open set with probability 1 the tangent measure isd scale invariant whered is the similarity dimension of moreover we approximate the tangential distribution by means of and use this in order to prove that the hausdorff dimension of the tangent measure equalsd since the former coincides with the hausdorff dimension of we obtain an earlier result of mauldin and williams 9 as a corollary

on the optimal control of strongly nonlinear evolution equations abstract in this paper we study optimal control problems governed by nonlinear evolution equations with nonmonotone nonlinearities in the state first we prove an existence theorem then using the penalty method we obtain necessary conditions for optimality from them we derive a bang bang characterization of optimal controls finally an example of a nonlinear parabolic optimal control system is worked out in detail

thick points of super brownian motion we determine for a super brownian motion xt t 0 in d d 3 the precise gauge function such that almost surely on survival up to time t open image in new window improving a result of barlow evans and perkins about the most visited sites of super brownian motion we also determine upper and lower bounds for the hausdorff dimension spectrum of thick points refining the multifractal analysis of super brownian motion by taylor and perkins the upper bound conjectured to be sharp involves a constant which can be characterized in terms of the upper tails of the associated equilibrium palm distribution