fourier transform of self affine measures abstract suppose f is a self affine set on r d d 2 which is not a singleton associated to affine contractions f j a j b j a j gl d r b j r d j a for some finite a we prove that if the group generated by the matrices a j j a forms a proximal and totally irreducible subgroup of gl d r then any self affine measure p j f j p j 1 0 p j 1 j a on f is a rajchman measure the fourier transform 0 as as an application this shows that self affine sets with proximal and totally irreducible linear parts are sets of rectangular multiplicity for multiple trigonometric series moreover if the zariski closure of is connected real split lie group in the zariski topology then has a power decay at infinity hence is l p improving for all 1 p and f has positive fourier dimension in dimension d 2 3 the irreducibility of and non compactness of the image of in pgl d r is enough for power decay of the proof is based on quantitative renewal theorems for random walks on the sphere s d 1

dimension of slices through the sierpinski carpet for lebesgue typical the intersection of the sierpinski carpet with a line has if non empty dimension where fix the slope then we shall show on the one hand that this dimension is strictly less than for lebesgue almost every on the other hand for almost every according to the angle projection of the natural measure on this dimension is at least for any we find a connection between the box dimension of this intersection and the local dimension of at

thick points for planar brownian motion and the erdős taylor conjecture on random walk and conjectured that the limit exists and equals 1 tr a s the importance of determining the value of this limit is clarified in 1 3 below where this value appears in the power laws governing the local time of the walk the erd6s taylor conjecture was quoted in the book by r v sz 19 w but to the best of our knowledge the bounds in 1 1 were not improved prior to the present paper as it turns out an important step towards our solution of the erd6s taylor conjecture was the formulation by perkins and taylor 17 of an analogous problem on the maximal occupation measure that planar brownian motion run for unit time can

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

a dimension formula for bernoulli convolutions we present a dynamical approach to the study of the singularity of infinitely convolved bernoulli measuresv for the golden section we introducev as the transverse measure of the maximum entropy measure on the repelling set invariant for contracting maps of the square the fat baker s transformation our approach strongly relies on the markov structure of the underlying dynamical system indeed if golden mean the fat baker s transformation has a very simple markov coding the ambiguity of order two of this coding which appears when projecting on the line due to passages for the central overlapping zone can be expressed by means of products of matrices of order two this product has a markov distribution inherited by the markov structure of the map the dimension of the projected measure is therefore associated to the growth of this product our dimension formula appears in a natural way as a version of the furstenberg guivarch formula our technique provides an explicit dimension formula and most important provides a formalism well suited for the multifractal analysis of this measure as we will show in a forthcoming paper

tensor network states and geometry tensor network states are used to approximate ground states of local hamiltonians on a lattice in d spatial dimensions different types of tensor network states can be seen to generate different geometries matrix product states mps in d 1 dimensions as well as projected entangled pair states peps in d 1 dimensions reproduce the d dimensional physical geometry of the lattice model in contrast the multi scale entanglement renormalization ansatz mera generates a d 1 dimensional holographic geometry here we focus on homogeneous tensor networks where all the tensors in the network are copies of the same tensor and argue that certain structural properties of the resulting many body states are preconditioned by the geometry of the tensor network and are therefore largely independent of the choice of variational parameters indeed the asymptotic decay of correlations in homogeneous mps and mera for d 1 systems is seen to be determined by the structure of geodesics in the physical and holographic geometries respectively whereas the asymptotic scaling of entanglement entropy is seen to always obey a simple boundary law that is again in the relevant geometry this geometrical interpretation offers a simple and unifying framework to understand the structural properties of and helps clarify the relation between different tensor network states in addition it has recently motivated the branching mera a generalization of the mera capable of reproducing violations of the entropic boundary law in d 1 dimensions

on equality of hausdorff and affinity dimensions via self affine measures on positive subsystems under mild conditions we show that the affinity dimension of a planar self affine set is equal to the supremum of the lyapunov dimensions of self affine measures supported on self affine proper subsets of the original set these self affine subsets may be chosen so as to have stronger separation properties and in such a way that the linear parts of their affinities are positive matrices combining this result with some recent breakthroughs in the study of self affine measures and their associated furstenberg measures we obtain new criteria under which the hausdorff dimension of a self affine set equals its affinity dimension for example applying recent results of barany hochman solomyak and rapaport we provide many new explicit examples of self affine sets whose hausdorff dimension equals its affinity dimension and for which the linear parts do not satisfy any positivity or domination assumptions

random countable iterated function systems with overlaps and applications abstract we study invariant measures for random countable finite or infinite conformal iterated function systems ifs with arbitrary overlaps we do not assume any type of separation condition we prove under a mild assumption of finite entropy the dimensional exactness of the projections of invariant measures from the shift space and we give a formula for their dimension in the context of random infinite conformal iterated function systems with overlaps there exist many differences between our case and the finite deterministic case studied in 7 and we introduce new methods specific to the infinite and random case we apply our results towards a problem related to a conjecture of lyons about random continued fractions 11 and show that for all parameters 0 the invariant measure is exact dimensional and in addition we give estimates for the pointwise and hausdorff dimension of for in a certain interval the finite ifs determining these continued fractions is not hyperbolic but we can associate to it a random infinite ifs of contractions which have overlaps we study then also other large classes of random countable iterated function systems with overlaps namely a several types of random iterated function systems related to kahane salem sets and b randomized infinite ifs in the plane which have uniformly bounded number of disc overlaps for all the above classes we prove dimensional exactness and we find lower and upper estimates for the pointwise and hausdorff packing dimensions of the projection measures

on homogenization and scaling limit of some gradient perturbations of a massless free field we study the continuum scaling limit of some statistical mechanical models defined by convex hamiltonians which are gradient perturbations of a massless free field by proving a central limit theorem for these models we show that their long distance behavior is identical to a new homogenized continuum massless free field we shall also obtain some new bounds on the 2 point correlation functions of these models

a self similar measure with dense rotations singular projections and discrete slices abstract we construct a planar homogeneous self similar measure with strong separation dense rotations and dimension greater than 1 such that there exist lines for which dimension conservation does not hold and the projection of the measure is singular in fact the set of such directions is residual and the typical slices of the measure perpendicular to these directions are discrete

the beale kato majda criterion for the 3d magneto hydrodynamics equations we study the blow up criterion of smooth solutions to the 3d mhd equations by means of the littlewood paley decomposition we prove a beale kato majda type blow up criterion of smooth solutions via the vorticity of velocity only namely sup j in mathbb z int 0 t delta j nabla times u infty dt where j is the frequency localization operator in the littlewood paley decomposition

tightness of the recentered maximum of the two dimensional discrete gaussian free field we consider the maximum of the discrete two dimensional gaussian free field gff in a box and prove that its maximum centered at its mean is tight settling a longstanding conjecture the proof combines a recent observation by bolthausen deuschel and zeitouni with elements from bramson s results on branching brownian motion and comparison theorems for gaussian fields an essential part of the argument is the precise evaluation up to an error of order 1 of the expected value of the maximum of the gff in a box related gaussian fields such as the gff on a two dimensional torus are also discussed 2011 wiley periodicals inc

equilibrium states for factor maps between subshifts abstract let x y be a factor map where x x and y y are subshifts over finite alphabets assume that x satisfies weak specification let a a 1 a 2 r 2 with a 1 0 and a 2 0 let f be a continuous function on x with sufficient regularity holder continuity for instance we show that there is a unique shift invariant measure on x that maximizes f d a 1 h x a 2 h 1 y in particular taking f 0 we see that there is a unique invariant measure on x that maximizes the weighted entropy a 1 h x a 2 h 1 y which answers an open question raised by gatzouras and peres 1996 in 15 an extension is given to high dimensional cases as an application we show that for each compact invariant set k on the k torus under a diagonal endomorphism if the symbolic coding of k satisfies weak specification then there is a unique invariant measure supported on k so that dim h dim h k

hausdorff dimension of the level sets of self affine functions abstract given a partition of 0 1 by closed intervals i 1 i k 0 1 with k 2 given 0 1 and closed intervals j i i 1 k with i j j j given 1 k 0 1 let 0 1 0 1 be the compact set satisfying that i 1 k i i 1 j i i where for an interval i a b 0 1 and 1 1 i 0 1 i is the linear map such that i 1 0 a i 1 1 b and i 1 0 b i 1 1 a such is a graph of a borel function f almost surely and is called a self affine set of function type we obtain the hausdorff dimension of the level set y x x y in the case that f 1 has a bounded density with respect to where is the lebesgue measure on 0 1

thick points of the gaussian free field let u c be a bounded domain with smooth boundary and let f be an instance of the continuum gaussian free field on u with respect to the dirichlet inner product u f x g x dx the set t a u of a thick points of f consists of those z u such that the average of f on a disk of radius r centered at z has growth a log 1 r as r 0 we show that for each 0 a 2 the hausdorff dimension of t a u is almost surely 2 a that 2 a t a u when 0 2 furthermore we prove that t a u is invariant under conformal transformations in an appropriate sense the notion of a thick point is connected to the liouville quantum gravity measure with parameter y given formally by dz e 2 f z dz considered by duplantier and sheffield

hausdorff dimension of planar self affine sets and measures let x bigcup varphi i x be a strongly separated self affine set in mathbb r 2 or one satisfying the strong open set condition under mild non conformality and irreducibility assumptions on the matrix parts of the varphi i we prove that dim x is equal to the affinity dimension and similarly for self affine measures and the lyapunov dimension the proof is via analysis of the dimension of the orthogonal projections of the measures and relies on additive combinatorics methods

density matrix formulation for quantum renormalization groups a generalization of the numerical renormalization group procedure used first by wilson for the kondo problem is presented it is shown that this formulation is optimal in a certain sense as a demonstration of the effectiveness of this approach results from numerical real space renormalization group calculations for heisenberg chains are presented

dimension and product structure of hyperbolic measures we prove that every hyperbolic measure invariant under a c 1 fi difieomorphism of a smooth riemannian manifold possesses asymptotically almost local product structure i e its density can be approximated by the product of the densities on stable and unstable manifolds up to small exponentials this has not been known even for measures supported on locally maximal hyperbolic sets using this property of hyperbolic measures we prove the long standing eckmann ruelle conjecture in dimension theory of smooth dynamical systems the pointwise dimension of every hyperbolic measure invariant under a c 1 fi difieomorphism exists almost everywhere this implies the crucial fact that virtually all the characteristics of dimension type of the measure including the hausdorfi dimension box dimension and information dimension coincide this provides the rigorous mathematical justiflcation of the concept of fractal dimension for hyperbolic measures

local dimensions of measures on infinitely generated self affine sets abstract we show the existence of the local dimension of an invariant probability measure on an infinitely generated self affine set for almost all translations this implies that an ergodic probability measure is exactly dimensional furthermore the local dimension equals the minimum of the local lyapunov dimension and the dimension of the space

connecting the von neumann and renyi entropies for fermions we explore the relation between the von neumann entropy and the renyi entropies of integer orders for shift invariant quasi free fermionic lattice systems we investigate the approximation of the von neumann entropy by a combination of integer order renyi entropies and give an estimate for the quality of such an approximation

stability and convergence of semi implicit time stepping algorithm for stationary incompressible magnetohydrodynamics abstract for the stationary incompressible magnetohydrodynamics mhd equations we provide a new uniqueness assumption a0 and show the exponential stability of the solution then the semi implicit time stepping algorithm is used to solve the stationary mhd equations the algorithm is proved to be unconditionally stable the discrete velocity and magnetic field are bounded in l 0 l for any space and time step sizes the error estimates for the algorithm are deduced under the uniqueness conditions finally numerical experiments are performed to testify our theoretical analysis

some characterizations of domination we show that a cocycle has a dominated splitting if and only if there is a uniform exponential gap between singular values of its iterates then we consider sets in rm gl d mathbb r with the property that any cocycle with values in has a dominated splitting we characterize these sets in terms of existence of invariant multicones thus extending a two dimensional result by avila bochi and yoccoz we give an example showing how these multicones can fail to have convexity properties

absolute continuity of non homogeneous self similar measures abstract we prove that self similar measures on the real line are absolutely continuous for almost all parameters in the super critical region in particular confirming a conjecture of s m ngai and y wang while recently there has been much progress in understanding absolute continuity for homogeneous self similar measures this is the first improvement over the classical transversality method in the general non homogeneous case in the course of the proof we establish new results on the dimension and fourier decay of a class of random self similar measures

on the regularity of weak solutions to the magnetohydrodynamic equations abstract in this paper we study the regularity of weak solution to the incompressible magnetohydrodynamic equations we obtain some sufficient conditions for regularity of weak solutions to the magnetohydrodynamic equations which is similar to that of incompressible navier stokes equations moreover our results demonstrate that the velocity field of the fluid plays a more dominant role than the magnetic field does on the regularity of solution to the magneto hydrodynamic equations

ledrappier young formula and exact dimensionality of self affine measures in this paper we solve the long standing open problem on exact dimensionality of self affine measures on the plane we show that every self affine measure on the plane is exact dimensional regardless of the choice of the defining iterated function system in higher dimensions under certain assumptions we prove that self affine and quasi self affine measures are exact dimensional in both cases the measures satisfy the ledrappier young formula

the metric entropy of diffeomorphisms part ii relations between entropy exponents and dimension on considere f m m m m un c 2 diffeomorphisme f d une variete de riemann compacte m preservant une mesure de probabilite de borel m soit hm f l entropie metrique de f et 1 x r x x les exposants de lyapunov distincts en x on etablit une formule qui relie h m f et les r x x

on the dimension of furstenberg measure for sl _ 2 mathbb r random matrix products let mu be a measure on sl 2 mathbb r generating a non compact and totally irreducible subgroup and let nu be the associated stationary furstenberg measure for the action on the projective line we prove that if mu is supported on finitely many matrices with algebraic entries then begin aligned dim nu min big 1 frac h mathrm mathrm rw mu 2 chi mu big end aligned where h mathrm mathrm rw mu is the random walk entropy of mu chi mu is the lyapunov exponent for the random matrix product associated with mu and dim denotes pointwise dimension in particular for every delta 0 there is a neighborhood u of the identity in sl 2 mathbb r such that if a measure mu in mathcal p u is supported on algebraic matrices with all atoms of size at least delta and generates a group which is non compact and totally irreducible then its stationary measure nu satisfies dim nu 1

regularity criterion for 3d mhd fluid passing through the porous medium in terms of gradient pressure abstract the aim of the paper is to establish regularity criteria for the weak solution of fluid passing through the porous media in r 3 our results show that if p l with 2 3 3 1 then the weak solution is regular and unique if p l with 2 3 2 3 2 then the weak solution is regular and unique

solutions for semilinear parabolic equations in lp and regularity of weak solutions of the navier stokes system abstract we construct a unique local regular solution in l q 0 t l p for a class of semilinear parabolic equations which includes the semilinear heat equation u t u u u 0 and the navier stokes system here p and q are so chosen that the norm of l q 0 t l p is dimensionless or scaling invariant the main relation between p and q for the semilinear heat equation is 1 q 1 p 1 q n 2 provided that initial data are in l r with r n 2 1 where n is the space dimension applying our regular solutions to the navier stokes system we show that the k 2 dimensional hausdorff measure of possible time singularities of a turbulent solution is zero if the turbulent solution is in l q 0 t l p where k 2 q nq p p n 1 q we show moreover that a turbulent solution is regular if it is in c 0 t l n

the metric entropy of diffeomorphisms part i characterization of measures satisfying pesin s entropy formula soit m une variete de riemann compacte soit f m m un diffeomorphisme et soit m une mesure de probabilite de borel f invariante sur m on identifie les mesures pour lesquelles l inegalite de margulis et ruelle qui relie l entropie aux exposants de lyapunov atteint l egalite on demontre une formule valable pour toutes les mesures invariantes

the hausdorff and dynamical dimensions of self affine sponges a dimension gap result we construct a self affine sponge in mathbb r 3 whose dynamical dimension i e the supremum of the hausdorff dimensions of its invariant measures is strictly less than its hausdorff dimension this resolves a long standing open problem in the dimension theory of dynamical systems namely whether every expanding repeller has an ergodic invariant measure of full hausdorff dimension more generally we compute the hausdorff and dynamical dimensions of a large class of self affine sponges a problem that previous techniques could only solve in two dimensions the hausdorff and dynamical dimensions depend continuously on the iterated function system defining the sponge implying that sponges with a dimension gap represent a nonempty open subset of the parameter space

on multifractal formalism for self similar measures with overlaps let mu be a self similar measure generated by an ifs varphi phi i i 1 ell of similarities on mathbb r d d ge 1 when varphi is dimensional regular see definition 1 1 we give an explicit formula for the l q spectrum tau mu q of mu over 0 1 and show that tau mu is differentiable over 0 1 and the multifractal formalism holds for mu at any alpha in tau mu 1 tau mu 0 we also verify the validity of the multifractal formalism of mu over tau mu infty tau mu 0 for two new classes of overlapping algebraic ifss by showing that the asymptotically weak separation condition holds for one of them the proof appeals to the recent result of shmerkin ann math 2 189 2 319 391 2019 on the l q spectrum of self similar measures

multifractal formalism for almost all self affine measures we conduct the multifractal analysis of self affine measures for almost all family of affine maps besides partially extending falconer s formula of lq spectrum outside the range 1 q 2 the multifractal formalism is also partially verified

local time of self affine sets of brownian motion type revisited abstract in 5 we studied self affine sets of brownian motion type we obtained the image measure induced by the lebesgue measure on the domain space if it is absolutely continuous the density function is called the local time the local times are obtained by solving so called jigsaw puzzle in this paper we are specially interested in the self affine set with 3 pieces with the orientations up down up see the picture below

some mathematical questions related to the mhd equations some questions relating to the large time behavior of the solutions of mhd equations for a viscous incompressible resistive fluid are investigated the physical system is briefly described and the functional setting of the equations a flow in a bounded domain or in whole space with a space periodicity property in all directions the main existence and uniqueness results for weak and strong solutions of the mhd equations are recalled regularity properties and bounds on the solutions to the equations which are valid for all time are established and the concept of functional invariant sets is introduced which is contained in the space of smooth functions if the data are sufficiently regular the squeezing property of the trajectories are stated and it is shown that any functional invariant set for the mhd equations and in particular any attractor has a finite haussdorf dimension the flow is found to be totally determined for large dimensions by a finite number of parameters 26 references

quantum spin chain toeplitz determinants and the fisher hartwig conjecture we consider the one dimensional quantum spin chain which is called the xx model xx0 model or isotropic xy model in a transverse magnetic field we are mainly interested in the entropy of a block of lneighboring spins at zero temperature and of an infinite system we represent the entropy in terms of a toeplitz determinant and calculate the asymptotic analytically we derive the first two terms of the asymptotic decomposition interestingly these two terms of decomposition clearly show a length scale related to the field h

finitely correlated states on quantum spin chains we study a construction that yields a class of translation invariant states on quantum spin chains characterized by the property that the correlations across any bond can be modeled on a finite dimensional vector space these states can be considered as generalized valence bond states and they are dense in the set of all translation invariant states we develop a complete theory of the ergodic decomposition of such states including the decomposition into periodic neel ordered states the ergodic components have exponential decay of correlations all states considered can be obtained as local functions of states of a special kind so called purely generated states which are shown to be ground states for suitably chosen finite range vbs interactions we show that all these generalized vbs models have a spectral gap our theory does not require symmetry of the state with respect to a local gauge group in particular we illustrate our results with a one parameter family of examples which are not isotropic except for one special case this isotropic model coincides with the one dimensional antiferromagnet recently studied by affleck kennedy lieb and tasaki

local time of self affine sets of brownian motion type and the jigsaw puzzle problem abstract let 0 1 0 1 be the solution of the set equation i 1 k i i 1 j i i where for an interval i a b 0 1 and 1 1 i 0 1 i is the linear map such that i 1 0 a i 1 1 b i 1 0 b i 1 1 a and i i i 1 k is a partition of 0 1 with j i i i 1 2 thus is a graph of a borel function f almost surely and it is called a self affine set of brownian motion type let be the lebesgue measure on 0 1 and let f 1 the density d d if it exists is called the local time of and it has been studied it is known that dim h 3 2 if exists in the present study is obtained by solving the so called jigsaw puzzle on j i i i 1 k i e the problem of decomposing into a sum of its self similar images with the support j i and the orientation i for i 1 k

extremes of the discrete two dimensional gaussian free field we consider the lattice version of the free field in two dimensions and study the fractal structure of the sets where the field is unusually high or low we then extend some of our computations to the case of the free field conditioned on being everywhere nonnegative for example we compute the width of the largest downward spike of a given length through the prism of these results we find that the extrema of the free field under entropic repulsion minus its mean and those of the unconditioned free field are identical finally when compared to previous results these findings reveal a suggestive analogy between the square of the free field and the two dimensional simple random walk on the discrete torus

tensor network methods for invariant theory invariant theory is concerned with functions that do not change under the action of a given group here we communicate an approach based on tensor networks to represent polynomial local unitary invariants of quantum states this graphical approach provides an alternative to the polynomial equations that describe invariants which often contain a large number of terms with coefficients raised to high powers this approach also enables one to use known methods from tensor network theory such as the matrix product state mps factorization when studying polynomial invariants as our main example we consider invariants of mpss we generate a family of tensor contractions resulting in a complete set of local unitary invariants that can be used to express the r nyi entropies we find that the graphical approach to representing invariants can provide structural insight into the invariants being contracted as well as an alternative and sometimes much simpler means to study polynomial invariants of quantum states in addition many tensor network methods such as mpss contain excellent tools that can be applied in the study of invariants

entropic repulsion and the maximum of the two dimensional harmonic crystal we consider the lattice version of the free field in two dimensions also called harmonic crystal the main aim of the paper is to discuss quantitatively the entropic repulsion of the random surface in the presence of a hard wall the basic ingredient of the proof is the analysis of the maximum of the field which requires a multiscale analysis reducing the problem essentially to a problem on a field with a tree structure

hausdorff dimension for randomly perturbed self affine attractors in this paper we shall consider a self affine iterated function system in mathbb r d d 2 where we allow a small random translation at each application of the contractions we compute the dimension of a typical attractor of the resulting random iterated function system complementing a famous deterministic result of falconer which necessarily requires restrictions on the norms of the contraction however our result has the advantage that we do not need to impose any additional assumptions on the norms this is of benefit in practical applications where such perturbations would correspond to the effect of random noise we also give analogous results for the dimension of ergodic measures in terms of their lyapunov dimension finally we apply our method to a problem originating in the theory of fractal image compression

multifractal measures and a weak separation condition abstract we define a new separation property on the family of contractive similitudes that allows certain overlappings this property is weaker than the open set condition of hutchinson it includes the well known class of infinite bernoulli convolutions associated with the p v numbers and the solutions of the two scale dilation equations our main purpose in this paper is to prove the multifractal formalism under such condition

on the complexity of the set of codings for self similar sets and a variation on the construction of champernowne abstract let f p 0 p n be a collection of points in r d the set f naturally gives rise to a family of iterated function systems consisting of contractions of the form s i x x 1 p i i 0 n where 0 1 and x r d given f and it is well known that there exists a unique non empty compact set x satisfying x i 0 n s i x for each x x there exists a sequence a j j 1 0 n n satisfying x lim j s a 1 s a j 0 we call such a sequence a coding of x in this paper we prove that for any f and k n there exists k f 0 such that if 1 k f 1 then every point in the interior of x has a coding which is k simply normal similarly we prove that there exists u n i f 0 such that if 1 u n i f 1 then every point in the interior of x has a coding containing all finite words for some specific choices of f we obtain lower bounds for k f and u n i f we also prove some weaker statements that hold in the more general setting when the similarities in our iterated function systems exhibit different rates of contraction our proofs rely on a variation of a well known construction of a normal number due to champernowne and an approach introduced by erd s and komornik

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

tensor network contractions for sat the computational cost of counting the number of solutions satisfying a boolean formula which is a problem instance of sat has proven subtle to quantify even when finding individual satisfying solutions is computationally easy e g 2 sat which is in mathsf p determining the number of solutions can be mathsf p hard recently computational methods simulating quantum systems experienced advancements due to the development of tensor network algorithms and associated quantum physics inspired techniques by these methods we give an algorithm using an axiomatic tensor contraction language for n variable sat instances with complexity o g cd o 1 2 c where c is the number of copy tensors g is the number of gates and d is the maximal degree of any copy tensor thus n variable counting problems can be solved efficiently when their tensor network expression has at most o log n copy tensors and polynomial fan out this framework also admits an intuitive proof of a variant of the tovey conjecture the r 1 sat instance of the dubois tovey theorem this study increases the theory expressiveness and application of tensor based algorithmic tools and provides an alternative insight on these problems which have a long history in statistical physics and computer science

log gases on quadratic lattices via discrete loop equations and q boxed plane partitions abstract we study a general class of log gas ensembles on shifted quadratic lattices we prove that the corresponding empirical measures satisfy a law of large numbers and that their global fluctuations are gaussian with a universal covariance we apply our general results to analyze the asymptotic behavior of a q boxed plane partition model introduced by borodin gorin and rains in particular we show that the global fluctuations of the height function on a fixed slice are described by a one dimensional section of a pullback of the two dimensional gaussian free field our approach is based on a q analogue of the schwinger dyson or loop equations which originate in the work of nekrasov and his collaborators and extends the methods developed by borodin gorin and guionnet to quadratic lattices

the construction of the d 1 dimensional gaussian droplet the aim of this note is to study the asymptotic behavior of a gaussian random field under the condition that the variables are positive and the total volume under the variables converges to some fixed numberv 0 in the context of statistical mechanics this corresponds to the problem of constructing a droplet on a hard wall with a given volume we show that properly rescaled the profile of a gaussian configuration converges to a smooth hypersurface which solves a quadratic variational problem our main tool is a scaling dependent large deviation principle for random hypersurfaces

the limited rademacher functions and bernoulli convolutions associated with pisot numbers abstract in this paper we give a systematical study of the local structures and fractal indices of the limited rademacher functions and bernoulli convolutions associated with pisot numbers for a given pisot number in the interval 1 2 we construct a finite family of non negative matrices maybe non square such that the corresponding fractal indices can be re expressed as some limits in terms of products of these non negative matrices we are especially interested in the case that the associated pisot number is a simple pisot number i e the unique positive root of the polynomial x k x k 1 x 1 k 2 3 in this case the corresponding products of matrices can be decomposed into the products of scalars based on which the precise formulas of fractal indices as well as the multifractal formalism are obtained

dominos and the gaussian free field we define a scaling limit of the height function on the domino tiling model dimer model on simply connected regions in z 2 and show that it is the massless free field a gaussian process with independent coefficients when expanded in the eigenbasis of the laplacian

smoothness of projections bernoulli convolutions and the dimension of exceptions erd s 1939 1940 studied the distribution of the random series p 0 n and showed that is singular for infinitely many 1 2 1 and absolutely continuous for a e in a small interval 1 1 solomyak 1995 proved a conjecture made by garsia 1962 that is absolutely continuous for a e 1 2 1 in order to sharpen this result we have developed a general method that can be used to estimate the hausdorff dimension of exceptional parameters in several contexts in particular we prove for any 0 1 2 the set of 0 1 such that is singular has hausdorff dimension strictly less than 1 for any borel set a rd with hausdorff dimension dim a d 1 2 there are points x a such that the pinned distance set x y y a has positive lebesgue measure moreover the set of x where this fails has hausdorff dimension at most d 1 dim a let k denote the middle cantor set for 1 2 and let k r be any compact set peres and solomyak 1998 showed that for a e 0 1 2 such that dim k dimk 1 the sum k k has positive length we show that the set of exceptional in this statement has hausdorff dimension at most 2 dim k dimk 0 for any borel set e rd with dim e 2 almost all orthogonal projections of e onto lines through the origin have nonempty interior and the exceptional set of lines where this fails has dimension at most d 1 dim e if is a borel probability measure on rd with correlation dimension greater than m 2 then for a prevalent set of c1 maps f rd rm in the sense described by hunt sauer and yorke 1992 the image of under f has a density with at least fractional derivatives in l2 rm

hausdorff dimension of the limit sets of some planar geometric constructions abstract we determine the hausdorff and box dimension of the limit sets for some class of planar non moran like geometric constructions generalizing the bedford mcmullen general sierpinski carpets the class includes affine constructions generated by an arbitrary partition of the unit square by a finite number of horizontal and vertical lines as well as some non affine examples e g the flexed sierpinski gasket

dimensions of invariant sets of expanding maps we consider a compact invariant set of an expanding map of a manifoldm and give upper and lowerbounds for the hausdorff dimension dimh and box dimensionsdimb and dimb these bounds are given in terms of the topological entropy topological pressure and uniform lyapunov exponents of the map

thick points for spatial brownian motion multifractal analysis of occupation measure let t x r denote the total occupation measure of the ball of radius r centered at x for brownian motion in ir we prove that sup x 1 t x r r log r 16 a s as r 0 thus solving a problem posed by taylor in 1974 furthermore for any a 0 16 the hausdorff dimension of the set of thick points x for which lim sup r 0 t x r r log r a is almost surely 2 a 8 this is the correct scaling to obtain a nondegenerate multifractal spectrum for brownian occupation measure analogous results hold for brownian motion in any dimension d 3 these results are related to the lil of ciesielski and taylor 1962 for the brownian occupation measure of small balls in the same way that levy s uniform modulus of continuity and the formula of orey and taylor 1974 for the dimension of fast points are related to the usual lil we also show that the lim inf scaling of t x r is quite different we exhibit non random c1 c2 0 such that c1 supx lim infr 0 t x r r 2 c2 a s in the course of our work we provide a general framework for obtaining lower bounds on the hausdorff dimension of random fractals of limsup type

leaf to leaf distances and their moments in finite and infinite ordered m ary tree graphs we study the leaf to leaf distances on full and complete m ary graphs using a recursive approach in our formulation leaves are ordered along a line we find explicit analytical formulae for the sum of all paths for arbitrary leaf to leaf distance r as well as the average path lengths and the moments thereof we show that the resulting explicit expressions can be recast in terms of hurwitz lerch transcendants results for periodic trees are also given for incomplete random binary trees we provide first results by numerical techniques we find a rapid drop of leaf to leaf distances for large r

tangent space gradient optimization of tensor network for machine learning the gradient based optimization method for deep machine learning models suffers from gradient vanishing and exploding problems particularly when the computational graph becomes deep in this work we propose the tangent space gradient optimization tsgo for probabilistic models to keep the gradients from vanishing or exploding the central idea is to guarantee the orthogonality between variational parameters and gradients the optimization is then implemented by rotating the parameter vector towards the direction of gradient we explain and test tsgo in tensor network tn machine learning where tn describes the joint probability distribution as a normalized state in hilbert space we show that the gradient can be restricted in tangent space of 1 hypersphere instead of additional adaptive methods to control the learning rate in deep learning the learning rate of tsgo is naturally determined by rotation angle as tan our numerical results reveal better convergence of tsgo in comparison to the off the shelf adam

dimension theory of iterated function systems let si l i 1 be an iterated function system ifs on d with attractor k let denote the one sided full shift over the alphabet 1 l we define the projection entropy function h on the space of invariant measures on e associated with the coding map k and develop some basic ergodic properties about it this concept turns out to be crucial in the study of dimensional properties of invariant measures on k we show that for any conformal ifs respectively the direct product of finitely many conformal ifss without any separation condition the projection of an ergodic measure under is always exactly dimensional and its hausdorff dimension can be represented as the ratio of its projection entropy to its lyapunov exponent respectively the linear combination of projection entropies associated with several coding maps furthermore for any conformal ifs and certain affine ifss we prove a variational principle between the hausdorff dimension of the attractors and that of projections of ergodic measures 2008 wiley periodicals inc

the law of large numbers for the maximum of almost gaussian log correlated fields coming from random matrices we compute the leading asymptotics as n rightarrow infty of the maximum of the field q n q log q a n q in mathbb c for any unitarily invariant hermitian random matrix a n associated to a non critical real analytic potential hence we verify the leading order in a conjecture of fyodorov and simm nonlinearity 29 2837 2016 arxiv 1503 07110 math ph formulated for the gue the method relies on a classical upper bound and a more sophisticated lower bound based on a variant of the second moment method which exploits the hyperbolic branching structure of the field q n q q in mathbb h specifically we compare q n to an idealized gaussian field by means of exponential moments in principle this method could also be applied to random fields coming from other point processes provided that one can compute certain mixed exponential moments for unitarily invariant ensembles we show that these assumptions follow from the fyodorov strahov formula fyodorov and strahov in j phys a 36 12 3203 3213 2003 https doi org 10 1088 0305 4470 36 12 320 and asymptotics of orthogonal polynomials derived in deift et al commun pure appl math 52 11 1335 1425 1999 https doi org 10 1002 sici 1097 0312 199911 52 11 3 0 co 2 1

structure of equilibrium states on self affine sets and strict monotonicity of affinity dimension a fundamental problem in the dimension theory of self affine sets is the construction of high dimensional measures which yield sharp lower bounds for the hausdorff dimension of the set a natural strategy for the construction of such high dimensional measures is to investigate measures of maximal lyapunov dimension these measures can be alternatively interpreted as equilibrium states of the singular value function introduced by falconer whilst the existence of these equilibrium states has been well known for some years their structure has remained elusive particularly in dimensions higher than two in this article we give a complete description of the equilibrium states of the singular value function in the three dimensional case showing in particular that all such equilibrium states must be fully supported in higher dimensions we also give a new sufficient condition for the uniqueness of these equilibrium states as a corollary giving a solution to a folklore open question in dimension three we prove that for a typical self affine set in r 3 removing one of the affine maps which defines the set results in a strict reduction of the hausdorff dimension