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

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

the minimal error method for the cauchy problem in linear elasticity numerical implementation for two dimensional homogeneous isotropic linear elasticity abstract in this paper yet another iterative procedure namely the minimal error method mem for solving stably the cauchy problem in linear elasticity is introduced and investigated furthermore this method is compared with another two iterative algorithms i e the conjugate gradient cgm and landweber fridman methods lfm previously proposed by marin et al marin l hao d n lesnic d 2002b conjugate gradient boundary element method for the cauchy problem in elasticity quarterly journal of mechanics and applied mathematics 55 227 247 and marin and lesnic marin l lesnic d 2005 boundary element landweber method for the cauchy problem in linear elasticity ima journal of applied mathematics 18 817 825 respectively in the case of two dimensional homogeneous isotropic linear elasticity the inverse problem analysed in this paper is regularized by providing an efficient stopping criterion that ceases the iterative process in order to retrieve stable numerical solutions the numerical implementation of the aforementioned iterative algorithms is realized by employing the boundary element method bem for two dimensional homogeneous isotropic linear elastic materials

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

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

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

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

minimum entropy control of closed loop tracking errors for dynamic stochastic systems the entropy has been used to characterize the uncertainty of the tracking error for general nonlinear and non gaussian stochastic systems a recursive optimization solution has been developed and the local stability condition of the closed loop system has been established the generality of this algorithm has been proved by the special case study of the minimum variance control for linear gaussian systems

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

observer based feedback controller design for a class of stochastic systems with non gaussian variables in this note the observer based feedback control problem is investigated for a class of non gaussian stochastic systems the proposed control strategy is the optimal control under a performance function measuring a sort of entropy of the state deviation sufficient conditions under which the closed loop system is locally and ultimately bounded in the mean square sense are presented simulation results demonstrate the effectiveness of the proposed design method

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

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

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

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

brief paper optimal probability density function control for narmax stochastic systems this paper presents a new control strategy for a class of non gaussian stochastic systems so that the output probability density function pdf of the system can be made to follow a desired pdf the system considered is represented by an nonlinear autoregressive and moving average with exogenous narmax inputs with input channel time delay and non gaussian noise a multi step ahead nonlinear cumulative cost function is used to improve tracking performance for this purpose a relationship between the pdfs of all the inputs and the pdfs of multiple step ahead output is formulated by constructing an auxiliary multivariate mapping by minimizing this performance function a new explicit predictive controller design algorithm is established with less conservatism than some previous results furthermore an improved approach is developed to guarantee the local stability of the closed loop system by tuning the weighting parameters recursively simulations are given to demonstrate the effectiveness of the proposed control algorithm and desired results have been obtained

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

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

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

brief paper feedback control of nonlinear stochastic systems for targeting a specified stationary probability density in the present paper an innovative procedure for designing the feedback control of multi degree of freedom mdof nonlinear stochastic systems to target a specified stationary probability density function spdf is proposed based on the technique for obtaining the exact stationary solutions of the dissipated hamiltonian systems first the control problem is formulated as a controlled dissipated hamiltonian system together with a target spdf then the controlled forces are split into a conservative part and a dissipative part the conservative control forces are designed to make the controlled system and the target spdf have the same hamiltonian structure mainly the integrability and resonance the dissipative control forces are determined so that the target spdf is the exact stationary solution of the controlled system five cases i e non integrable hamiltonian systems integrable and non resonant hamiltonian systems integrable and resonant hamiltonian systems partially integrable and non resonant hamiltonian systems and partially integrable and resonant hamiltonian systems are treated respectively a method for proving that the transient solution of the controlled system approaches the target spdf as t is introduced finally an example is given to illustrate the efficacy of the proposed design procedure

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

minimax optimal control of stochastic uncertain systems with relative entropy constraints this paper considers a new class of discrete time stochastic uncertain systems in which the uncertainty is described by a constraint on the relative entropy between a nominal noise distribution and the perturbed noise distribution this uncertainty description is a natural extension to the case of stochastic uncertain systems of the sum quadratic constraint uncertainty description this paper solves problems of worst case robust performance analysis and output feedback minimax optimal controller synthesis in a general nonlinear setting specializing these results to the linear case leads to a minimax linear quadratic gaussian lqg optimal controller this controller is defined by riccati difference equations and a kalman filter like state equation the paper also shows that the minimax lqg problem will have a solution if and only if a corresponding h sup spl infin control problem has a solution a linear example is presented to illustrate the minimax lqg methodology

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 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

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