on the invariant cantor sets of period doubling type of infinitely renormalizable area preserving maps since its inception in the 1970s at the hands of feigenbaum and independently coullet and tresser the study of renormalization operators in dynamics has been very successful at explaining univers

the high accurate block grid method for solving laplace s boundary value problem with singularities a high accurate difference analytical method is introduced for the solution of the mixed boundary value problem for laplace s equation on graduated polygons the polygon can have broken sections and be multiply connected the uniform estimate of the error of the approximate solution is of order o h6 whereas it is of order o h 6 r j p lambda j for the errors of p order derivatives p 1 2 in a finite neighborhood of reentry vertices here h is the mesh step rj is the distance from the current point to the vertex in question lambda j 1 a alpha j and a 1 or 2 depending on the types of boundary conditions further alpha j pi is the value of the interior angle at the considered vertex numerical experiments are illustrated in section ref sec8 to support the analysis made

on the hyperbolicity of the period doubling fixed point we give a new proof of the hyperbolicity of the fixed point for the period doubling renormalization operator using the local dynamics near a semi attractive fixed point in a banach space and the theory of holomorphic motions we also give a new proof of the exponential contraction of the feigenbaum renormalization operator in the hybrid class of the period doubling fixed point our proof uses the non existence of invariant line fields in the period doubling tower c mcmullen the topological convergence d sullivan and a new infinitesimal argument

rigidity of c 2 infinitely renormalizable unimodal maps given c 2 infinitely renormalizable unimodal maps f and g with a quadratic critical point and the same bounded combinatorial type we prove that they are c 1 conjugate along the closure of the corresponding forward orbits of the critical points for some 0

hausdorff dimension of julia sets of feigenbaum polynomials with high criticality we consider unimodal polynomials with feigenbaum topological type and critical points whose orders tend to infinity it is shown that the hyperbolic dimensions of their julia set go to 2 furthermore that the hausdorff dimensions of the basins of attraction of their feigenbaum attractors also tend to 2 the proof is based on constructing a limiting dynamics with a flat critical point

regular or stochastic dynamics in real analytic families of unimodal maps in this paper we prove that in any non trivial real analytic family of quasiquadratic maps almost any map is either regular i e it has an attracting cycle or stochastic i e it has an absolutely continuous invariant measure to this end we show that the space of analytic maps is foliated by codimension one analytic submanifolds hybrid classes this allows us to transfer the regular or stochastic property of the quadratic family to any non trivial real analytic family

dynamics and universality of unimodal mappings with infinite criticality we consider infinitely renormalizable unimodal mappings with topological type which are periodic under renormalization we study the limiting behavior of fixed points of the renormalization operator as the order of the critical point increases to infinity it is shown that a limiting dynamics exists with a critical point that is flat but still having a well behaved analytic continuation to a neighborhood of the real interval pinched at the critical point we study the dynamics of limiting maps and prove their rigidity in particular the sequence of fixed points of renormalization for finite criticalities converges uniformly on the real domain to a mapping of the limiting type

renormalizations and rigidity theory for circle homeomorphisms with singularities of the break type circle homeomorphisms with singularities of the break type are considered in the case when rotation numbers have periodic continued fraction expansion we establish hyperbolicity for renormalizations and then use it in order to prove the following rigidity result namely we show that any two homeomorphisms with a single break point are smoothly conjugate to each other provided they have the same quadratic irrational rotation number and the same size of a break

global hyperbolicity of renormalization for c r unimodal mappings in this paper we extend m lyubichi s recent results on the global hyperbolicity of renormalization of quadratic like germs to the space of cr unimodal maps with quadratic critical point we show that in this space the boundedtype limit sets of the renormalization operator have an invariant hyperbolic structure provided r i 2 a with a close to one as an intermediate step between lyubichi s results and ours we prove that the renormalization operator is hyperbolic in a banach space of real analytic maps we construct the local stable manifolds and prove that they form a continuous lamination whose leaves are c1 codimension one banach submanifolds of the ambient space and whose holonomy is c1 for some 0 we also prove that the global stable sets are c1 immersed codimension one submanifolds as well provided r i 3 a with a close to one as a corollary we deduce that in generic oneparameter families of cr unimodal maps the set of parameters corresponding to infinitely renormalizable maps of bounded combinatorial type is a cantor set with hausdorff dimension less than one

difference method of fourth order accuracy for the laplace equation with multilevel nonlocal conditions abstract we consider the multipoint nonlocal boundary value problem for the two dimensional laplace equation in a rectangular domain the solution of this problem is defined as a 9 point finite difference solution with the fourth order gluing operator of the local dirichlet boundary value problem by constructing a special method to find a function as the boundary value on the side of the rectangle where the nonlocal condition is given numerical experiments are illustrated to support the analysis made

common limits of fibonacci circle maps we show that limits for the critical exponent tending to exist in both critical circle homeomorphism of golden mean rotation number and fibonacci circle coverings moreover they are the same the limit map is not analytic at the critical point which is flat but has non trivial complex dynamics

radial basis function method for a multidimensional linear elliptic equation with nonlocal boundary conditions the development of numerical methods for the solution of partial differential equations pdes with nonlocal boundary conditions is important since such type of problems arise as mathematical models of various real world processes we use radial basis function rbf collocation technique for the solution of a multidimensional linear elliptic equation with classical dirichlet boundary condition and nonlocal integral conditions rbf based meshless methods are easily implemented and efficient especially for multidimensional problems formulated on complexly shaped domains in this paper properties of the method are investigated by studying two and three dimensional test examples with manufactured solutions we analyze the influence of the rbf shape parameter and the distribution of the nodes on the accuracy of the method as well as the influence of nonlocal conditions on the conditioning of the collocation matrix

spurious pressure in scott vogelius elements abstract we will analyze the characteristics of scott vogelius finite elements on singular vertices which cause spurious pressures on solving stokes equations a simple postprocessing will be suggested to remove those spurious pressures

persistence of the feigenbaum attractor in one parameter families consider a map 0 of class cr for large r of a manifold of dimension n greater than or equal to 2 having a feigenbaum attractor we prove that any such 0 is a point of a local codimension one manifold of cr transformations also exhibiting feigenbaum attractors in particular the attractor persists when perturbing a one parameter family transversal to that manifold at 0 we also construct such a transversal family for any given 0 and apply this construction to prove a conjecture by j palis stating that a map exhibiting a feigenbaum attractor can be perturbed to obtain homoclinic tangencies

complex bounds for real maps in this paper we prove complex bounds also referred to as a priori bounds for c 3 and in particular for analytic maps of the interval any c 3 mapping of the interval has an asymptotically holomorphic extension to a neighbourhood of the interval we associate to such a map a complex box mapping which provides a kind of markov structure for the dynamics moreover we prove universal geometric bounds on the shape of the domains and on the moduli between components of the range and domain such bounds show that the first return maps to these domains are well controlled and consequently such bounds form one of the corner stones in many recent results in one dimensional dynamics for example renormalization theory rigidity density of hyperbolicity and local connectivity of julia sets

quasiconformal homeomorphisms and dynamics i solution of the fatou julia problem on wandering domains on demontre un resultat dynamique sur l iteration d une application analytique complexe sur la sphere de riemann

the attractor of renormalization and rigidity of towers of critical circle maps we demonstrate the existence of a global attractor with a cantor set structure for the renormalization of critical circle mappings the set is invariant under a generalized renormalization transformation whose action on is conjugate to the two sided shift with a countable alphabet

on the existence of feigenbaum s fixed point we give a proof of the existence of ac2 even solution of feigenbaum s functional equation g text x lambda 0 1 g text g lambda 0 x g text 0 1 whereg is a map of 1 1 into itself it extends to a real analytic function over

on the hyperbolicity of lorenz renormalization we consider infinitely renormalizable lorenz maps with real critical exponent 1 of certain monotone combinatorial types we prove the existence of periodic points of the renormalization operator and that each map in the limit set of renormalization has an associated two dimensional strong unstable manifold for monotone families of lorenz maps we prove that each infinitely renormalizable combinatorial type has a unique representative within the family we also prove that each infinitely renormalizable map has no wandering intervals is ergodic and has a uniquely ergodic minimal cantor attractor of measure zero

finite difference methods for solution of nonlocal boundary value problems the article is devoted to the construction and investigation of various types of difference schemes relevant to nonlocal boundary value problems stated for multidimensional elliptic equations suitable tools for performing the investigation are developed including difference analogue of green s formula and inequalities corresponding to nonlocal boundary conditions

topological entropy of quadratic polynomials and dimension of sections of the mandelbrot set abstract a fundamental theme in holomorphic dynamics is that the local geometry of parameter space e g the mandelbrot set near a parameter reflects the geometry of the julia set hence ultimately the dynamical properties of the corresponding dynamical system we establish a new instance of this phenomenon in terms of entropy indeed we prove an entropy formula relating the entropy of a polynomial restricted to its hubbard tree to the hausdorff dimension of the set of rays landing on the corresponding vein in the mandelbrot set the results contribute to the recent program of w thurston of understanding the geometry of the mandelbrot set via the core entropy

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

renormalization in the henon family i universality but non rigidity in this paper geometric properties of infinitely renormalizable real henon like maps f in mathbb r 2 are studied it is shown that the appropriately defined renormalizations r n f converge exponentially to the one dimensional renormalization fixed point the convergence to one dimensional systems is at a super exponen tial rate controlled by the average jacobian and a universal function a x it is also shown that the attracting cantor set of such a map has hausdorff dimension less than 1 but contrary to the one dimensional intuition it is not rigid does not lie on a smooth curve and generically has unbounded geometry

combinatorics and topology of straightening maps i compactness and bijectivity abstract we study the parameter space structure of degree d 3 one complex variable polynomials as dynamical systems acting on c we introduce and study straightening maps these maps are a natural higher degree generalization of the ones introduced by douady and hubbard to prove the existence of small copies of the mandelbrot set inside itself we establish that straightening maps are always injective and that their image contains all the corresponding hyperbolic systems also we characterize straightening maps with compact domains moreover we give two classes of bijective straightening maps the first produces an infinite collection of embedded copies of the d 1 fold product of the mandelbrot set in the connectedness locus of degree d 3 the second produces an infinite collection of full families of quadratic connected filled julia sets in the cubic connectedness locus such that each filled julia set is quasiconformally embedded

on bitsadze samarskii type nonlocal boundary value problems for elliptic differential and difference equations well posedness the well posedness of the bitsadze samarskii type nonlocal boundary value problem in holder spaces with a weight is established the coercivity inequalities for the solution of the nonlocal boundary value problem for elliptic equations are obtained the stable second order of accuracy difference scheme for the approximate solution of the problem is presented the well posedness of this difference scheme in difference analogue of holder spaces is established for applications the almost coercivity and the coercivity estimates for solutions of difference schemes for elliptic equations are obtained

bounds on the unstable eigenvalue for period doubling bounds are given for the unstable eigenvalue of the period doubling operator for unimodal maps of the interval these bounds hold for all types of behaviour x r of the interval map near its critical point they are obtained by finding cones in function space which are invariant under the tangent map to the doubling operator at its fixed point

analyticity properties of the feigenbaum function analyticity properties of the feigenbaum function a solution ofg x 1g g x withg 0 1 g 0 0 g 0 0 are investigated by studying its inverse function which turns out to be herglotz or anti herglotz on all its sheets it is found thatg is analytic and uniform in a domain with a natural boundary

renormalization horseshoe for critical circle maps we prove that the invariant horseshoe for the renormalization of critical circle maps is globally uniformly hyperbolic with one dimensional unstable direction this extends our earlier result on hyperbolicity of periodic orbits of the renormalization and completes the proof of lanford s program

period doubling in maps with a maximum of order z abstract we consider the scaling behaviour in period doubling systems exemplified by the one dimensional map n 1 1 n z which has a maximum of order z z 1 the feigenbaum scaling factors and are studied as functions of z and more generally the scaling functions 1 and a in particular using the universal functions g x and h x we establish the inequality

on analytic perturbations of a family of feigenbaum like equations we prove existence of solutions 0 a of a family of feigenbaum like equations phi x 1 is an element of lambda phi phi lambda x is an element of x tau x 0 1 where epsilon is a small real number and tau is analytic and small on some complex neighborhood of 1 1 and real valued on r the family 0 1 appears in the context of period doubling renormalization for area preserving maps cf gaidashev and koch preprint 171 our proof is a development of ideas of h epstein cf epstein 1986 pi epstein 1988 in epstein 1989 4 adopted to deal with some significant complications that arise from the presence of the terms epsilon x tau x in eq 0 1 the method relies on a construction of novel a priori bounds for unimodal functions which turn out to be very tight we also obtain good bounds on the scaling parameter lambda a byproduct of the method is a new proof of the existence of a feigenbaum coullet tresser function

feigenbaum coullet tresser universality and milnor s hairiness conjecture we prove the feigenbaum coullet tresser conjecture on the hyperbolicity of the renormalization transformation of bounded type this gives the first computer free proof of the original feigenbaum observation of the universal parameter scaling laws we use the hyperbolicity theorem to prove milnor s conjectures on self similarity and hairiness of the mandelbrot set near the corresponding parameter values we also conclude that the set of real infinitely renormalizable quadratics of type bounded by some n 1 has hausdorff dimension strictly between 0 and 1 in the course of getting these results we supply the space of quadratic like germs with a complex analytic structure and demonstrate that the hybrid classes form a complex codimension one foliation of the connectedness locus

analysis of some finite elements for the stokes problem we study some finite elements which are used in the approximation of the stokes problem so as to obtain error estimates of optimal order resume nous etudions deux elements finis utilises pour l approximation du probleme de stokes et obtenons des estimations d erreur d ordre optimal

asymptotic and essentially singular solutions of the feigenbaum equation for suitably defined largen we express feigenbaum s equation as a singular schroder functional equation whose solution is obtained using a scaling ansatz in the limit of infiniten certain self consistency conditions on the scaled schroder solution lead to an essentially singular solution of feigenbaum s equation with a length scale factor of 0 0333 and a limiting feigenvalue of 30 50 in agreement with eckmann and wittwer s value of 0 0333831 and their conjectured estimate of 30

hausdorff dimension and conformal measures of feigenbaum julia sets 1 1 statement of the results one of the first questions usually asked about a fractal subset of r is whether it has the maximal possible hausdorff dimension n it certainly happens if the set has positive lebesgue measure on the other hand it is easy to construct fractal sets of zero measure but of dimension n moreover this phenomenon is often observable for fractal sets produced by conformal dynamical systems iterated rational functions or kleinian groups in particular the analogy with kleinian groups suggested that the julia sets of feigenbaum maps should have hausdorff dimension two in this paper we will show that this is not always the case a quadratic polynomial or more generally a quadratic like map is called feigenbaum if it is infinitely renormalizable of bounded combinatorial type with a priori bounds see 2 for a precise definition feigenbaum polynomials are remarkable dynamical systems whose geometry has been a focus of research for the past 25 years by analogy with kleinian groups it was anticipated that their julia sets have dimension 2 and this problem has been around for a while however in this paper we will show that this is not always the case

an asymptotic universal focal decomposition for non isochronous potentials galileo in the xvii century observed that the small os cillations of a pendulum seem to have constant period in fact the taylor expansion of the period map of the pendulum is constant up to second order in the initial angular velocity around the stable equi librium it is well known that for small oscillations of the pendulum and small intervals of time the dynamics of the pendulum can be ap proximated by the dynamics of the harmonic oscillator we study the dynamics of a family of mechanical systems that includes the pendulum at small neighbourhoods of the equilibrium but after long intervals of time so that the second order term of the period map can no longer be neglected we analyze such dynamical behaviour through a renor malization scheme acting on the dynamics of this family of mechanical systems the main theorem states that the asymptotic limit of this renormalization scheme is universal it is the same for all the elements in the considered class of mechanical systems as a consequence we obtain an universal asymptotic focal decomposition for this family of mechanical systems this paper is intended to be the first of a series of articles aiming at a semiclassical quantization of systems of the pendu lum type as a natural application of the focal decomposition associated to the two point boundary value problem

statistical properties of unimodal maps the quadratic family we prove that almost every nonregular real quadratic map is colleteckmann and has polynomial recurrence of the critical orbit proving a conjecture by sinai it follows that typical quadratic maps have excellent ergodic properties as exponential decay of correlations keller and nowicki young and stochastic stability in the strong sense baladi and viana this is an important step in achieving the same results for more general families of unimodal maps

stokes complexes and the construction of stable finite elements with pointwise mass conservation two families of conforming finite elements for the two dimensional stokes problem are developed guided by two discrete smoothed de rham complexes which we coin stokes complexes we show that the finite element pairs are inf sup stable and also provide pointwise mass conservation on very general triangular meshes

rigidity for infinitely renormalizable area preserving maps the period doubling cantor sets of strongly dissipative henon like maps with different average jacobian are not smoothly conjugated as was shown previously the jacobian rigidity conjecture says that the period doubling cantor sets of two dimensional henon like maps with the same average jacobian are smoothly conjugated this conjecture is true for average jacobian zero for example the one dimensional case the other extreme case is when the maps preserve area for example when the average jacobian is one indeed the main result presented here is that the period doubling cantor sets of area preserving maps in the universality class of the eckmann koch wittwer renormalization fixed point are smoothly conjugated

universality of critical circle covers for a class of critical circle covers we show that properly re scaled first return maps to a neighborhood of the critical point converge to universal limits for that to occur the critical point has to be sufficiently flat

puzzle geometry and rigidity the fibonacci cycle is hyperbolic we describe a new and robust method to prove rigidity results in complex dynamics the new ingredient is the geometry of the critical puzzle pieces under control of geometry and complex bounds two generalized polynomial like maps which admit a topological conjugacy quasiconformal outside the filled in julia set are indeed quasiconformally conjugate the proof uses a new abstract removability type result for quasiconformal maps following ideas of heinonen and koskela and of kallunki and koskela optimized for applications in complex dynamics we prove as the first application of this new method that for even criticalities distinct from two the period two cycle of the fibonacci renormalization operator is hyperbolic with dimensional unstable manifold

the boundary of chaos for interval mappings a goal in the study of dynamics on the interval is to understand the transition to positive topological entropy there is a conjecture from the 1980s that the only route to positive topological entropy is through a cascade of period doubling bifurcations we prove this conjecture in natural families of smooth interval maps and use it to study the structure of the boundary of mappings with positive entropy in particular we show that in families of mappings with a fixed number of critical points the boundary is locally connected and for analytic mappings that it is a cellular set

uniform rigidity sequences for weak mixing diffeomorphisms on d2 a and t2 abstract in the case of the disc d 2 the annulus a s 1 0 1 and the torus t 2 we will show that if a sequence of natural numbers satisfies a certain growth rate then there is a weak mixing diffeomorphism that is uniformly rigid with respect to that sequence the proof is based on a quantitative version of the anosov katok method with explicitly defined conjugation maps and the constructions are done in the c topology beyond that we can deduce a similar result in the real analytic topology in the case of t 2

on the metric properties of multimodal interval maps and c 2 density of axiom a in this paper we shall prove that axiom a maps are dense in the space of c 2 interval maps endowed with the c 2 topology as a step of the proof we shall prove real and complex a priori bounds for first return maps to certain small neighborhoods of the critical points of real analytic multimodal interval maps with non degenerate critical points we shall also discuss rigidity for interval maps without large bounds

on quasi conformal in compatibility of satellite copies of the mandelbrot set i douady and hubbard ann sci ec norm suppl 4 18 2 287 343 1985 introduced the notion of polynomial like maps they used it to identify homeomorphic copies mathbf m of the mandelbrot set inside the mandelbrot set mathbf m these copies can be primitive with a root cusp or satellite without a root cusp they conjectured that the primitive copies are quasiconformally homeomorphic to mathbf m and that the satellite ones are quasiconformally homeomorphic to mathbf m outside any small neighbourhood of the root these conjectures are now theorems due to lyubich ann math 149 319 420 1999 the satellite copies mathbf m p q are clearly not q c homeomorphic to mathbf m but are they mutually q c homeomorphic or even q c homeomorphic to half of the logistic mandelbrot set in this paper we prove that in general the induced douady hubbard homeomorphism is not the restriction of a q c homeomorphism for any two satellite copies mathbf m and mathbf m the induced douady hubbard homeomorphism is not q c if the root multipliers lambda text e 2 pi i p q and lambda text e 2 pi i p q have q not q

on the solution of a nonlocal problem in a rectangular domain we consider the bitsadze samarskii nonlocal boundary value problem for the two dimensional poisson equation the solution of this problem is defined as a solution of the local dirichlet boundary value problem by constructing a special method to find a function as the boundary value on the side of the rectangle where the nonlocal condition was given further the five point approximation of the laplace operator is used for the realization of the proposed method numerical experiments are illustrated in the last section to support the analysis made

divergence free finite elements on tetrahedral grids for k 6 it was shown two decades ago that the p k p k 1 mixed element on triangular grids approximating the velocity by the continuous p k piecewise polynomials and the pressure by the discontinuous p k 1 piecewise polynomials is stable for all k 4 provided the grids are free of a nearly singular vertex the problem with the method in 3d was posted then and remains open the problem is solved partially in this work it is shown that the p k p k 1 element is stable and of optimal order in approximation on a family of uniform tetrahedral grids for all k 6 the analysis is to be generalized to non uniform grids when we can deal with the complicity of 3d geometry for the divergence free elements the finite element spaces for the pressure can be avoided in computation if a classic iterated penalty method is applied the finite element solutions for the pressure are computed as byproducts from the iterate solutions for the velocity numerical tests are provided

a finite element method for elliptic equations on surfaces in this paper a new finite element approach for the discretization of elliptic partial differential equations on surfaces is treated the main idea is to use finite element spaces that are induced by triangulations of an outer domain to discretize the partial differential equation on the surface the method is particularly suitable for problems in which there is a coupling with a flow problem in an outer domain that contains the surface we give an analysis that shows that the method has optimal order of convergence both in the h 1 and in the l 2 norm results of numerical experiments are included that confirm this optimality

the periodic points of renormalization it will be shown that the renormalization operator acting on the space of smooth unimodal maps with critical exponent a 1 has periodic points of any combinatorial type a central question in the theory of dynamical systems is whether small scale geometrical properties of dynamical systems are determined by the combinatorial properties of the system indeed such universality of small scale geometry was discovered by coullet tresser and feigenbaum they studied infinitely renormalizable unimodal maps of the period doubling type and observed that the geometry of the invariant cantor set of such maps converges when looking at smaller and smaller scales furthermore they observed that the limiting geometry was universal in the sense that the small scale geometry of these cantor sets depends only on the local behavior of the map around the critical point this local behavior is specified by the critical exponent to explain the universality of geometry they defined the period doubling renormalization operator on a suitable space of unimodal maps this operator acts like a microscope the image under the renormalization operator is a unimodal map describing the geometry and dynamics on a smaller scale the universality of geometry was understood by conjecturing that the renormalization operator has a unique hyperbolic fixed point in particular the infinitely renormalizable unimodal maps form the stable manifold of the fixed point of the renormalization operator the first step in proving these conjectures is showing the existence of a renormalization fixed point

on the lebesgue measure of the feigenbaum julia set we show that the julia set of the feigenbaum polynomial has hausdorff dimension less than 2 and consequently it has zero lebesgue measure this solves a long standing open question