local connectivity of julia sets for unicritical polynomials we prove that the julia set j f of at most finitely renormalizable unicritical polynomial f z d c with all periodic points repelling is locally connected for d 2 it was proved by yoccoz around 1990 it follows from a priori bounds in a modified principal nest of puzzle pieces the proof of a priori bounds makes use of new analytic tools developed in kl09 that give control of moduli of annuli under maps of high degree

remote state estimation with communication costs for first order lti systems consider a first order linear time invariant discrete time system driven by process noise a pre processor that accepts causal measurements of the state of the system and a state estimator the pre processor and the state estimator are not co located and at every time step the pre processor transmits either a real number or a free symbol to the estimator we seek the pre processor and the estimator that jointly minimize a cost that combines two terms the expected squared state estimation error and a communication cost in our formulation the transmission of a real number from the pre processor to the estimator incurs a positive cost while free symbols induce zero cost this paper proves analytically that a symmetric threshold policy at the pre processor and a kalman like filter at the estimator which updates its estimate linearly in the presence of free symbols are jointly optimal for our problem

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

a quasi static discontinuous galerkin configurational force crack propagation method for brittle materials summary this paper presents a framework for r adaptive quasi static configurational force cf brittle crack propagation cast within a discontinuous galerkin dg symmetric interior penalty sipg finite element scheme cracks are propagated in discrete steps with a staggered algorithm along element interfaces which align themselves with the predicted crack propagation direction the key novelty of the work is the exploitation of the dg face stiffness terms existing along element interfaces to propagate a crack in a mesh independent r adaptive quasi static fashion driven by the cf at the crack tip this adds no new degrees of freedom to the data structure additionally as dg methods have element specific degrees of freedom a geometry driven p adaptive algorithm is also easily included allowing for more accurate solutions of the cf on a moving crack front further for non determinant systems we introduce an average boundary condition that restrains rigid body motion leading to a determinant system to the authors knowledge this is first time such a boundary condition has been described the proposed formulation is validated against single and multiple crack problems with single and mixed mode cracks demonstrating the predictive capabilities of the method this article is protected by copyright all rights reserved

adaptive sampling for linear state estimation when a sensor has continuous measurements but sends occasional messages over a data network to a supervisor which estimates the state the available packet rate fixes the achievable quality of state estimation when such rate limits turn stringent the sensor s messaging policy should be designed anew what are good causal messaging policies what should message packets contain what is the lowest possible distortion in a causal estimate at the supervisor is delta sampling better than periodic sampling we answer these questions for a markov state process under an idealized model of the network and the assumption of perfect state measurements at the sensor if the state is a scalar or a vector of low dimension then we can ignore sample quantization if in addition we can ignore jitter in the transmission delays over the network then our search for efficient messaging policies simplifies first each message packet should contain the value of the state at that time thus a bound on the number of data packets becomes a bound on the number of state samples second the remaining choice in messaging is entirely about the times when samples are taken for a scalar linear diffusion process we study the problem of choosing the causal sampling times that will give the lowest aggregate squared error distortion we stick to finite horizons and impose a hard upper bound n on the number of allowed samples we cast the design as a problem of choosing an optimal sequence of stopping times we reduce this to a nested sequence of problems each asking for a single optimal stopping time under an unproven but natural assumption about the least square estimate at the supervisor each of these single stopping problems are of standard form the optimal stopping times are random times when the estimation error exceeds designed envelopes for the case where the state is a brownian motion we give analytically the shape of the optimal sampling envelopes the shape of the envelopes under optimal delta sampling and their performances surprisingly we find that delta sampling performs badly hence when the rate constraint is a hard limit on the number of samples over a finite horizon we should not use delta sampling

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

discrete time controlled markov processes with average cost criterion a survey this work is a survey of the average cost control problem for discrete time markov processes the authors have attempted to put together a comprehensive account of the considerable research on this problem over the past three decades the exposition ranges from finite to borel state and action spaces and includes a variety of methodologies to find and characterize optimal policies the authors have included a brief historical perspective of the research efforts in this area and have compiled a substantial yet not exhaustive bibliography the authors have also identified several important questions that are still open to investigation

almost every real quadratic map is either regular or stochastic in this paper we complete a program to study measurable dynamics in the real quadratic family our goal was to prove that almost any real quadratic map pc z t x2 c c c 2 1 4 has either an attracting cycle or an absolutely continuous invariant measure the final step completed here is to prove that the set of infinitely renormalizable parametric values c c 2 1 4 has zero lebesgue measure we derive this from a renormalization theorem which asserts uniform hyperbolicity of the full renormalization operator this theorem gives the most general real version of the feigenbaum coullet tresser universality simultanuously for all combinatorial types

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

optimal communication scheduling and remote estimation over an additive noise channel abstract this paper considers a sequential sensor scheduling and remote estimation problem with one sensor and one estimator the sensor makes sequential observations about the state of an underlying memoryless stochastic process and makes a decision as to whether or not to send this measurement to the estimator the sensor and the estimator have the common objective of minimizing expected distortion in the estimation of the state of the process over a finite time horizon the sensor is either charged a cost for each transmission or constrained on transmission times as opposed to the prior work where communication between the sensor and the estimator was assumed to be perfect noiseless in this work an additive noise channel with fixed power constraint is considered hence the sensor has to encode its message before transmission under some technical assumptions we obtain the optimal encoding and estimation policies within the piecewise affine class in conjunction with the optimal transmission schedule the impact of the presence of a noisy channel is analyzed numerically based on dynamic programming this analysis yields some rather surprising results such as a phase transition phenomenon in the number of used transmission opportunities which was not encountered in the noiseless communication setting

erratum to real bounds ergodicity and negative schwarzian for multimodal maps here as before we define an open interval k to be nice if no iterate of k enters k this implies that if k1 and k2 are pullbacks of k then they are either disjoint or nested in lemma 9 page 762 it was implicitly assumed that v is disjoint from jn it is for this reason that the proof of theorem c 1 does not work unless we assume v is nice or something similar the proof of theorem c 1 as stated above is essentially the same as before using lemma 6 below instead of lemma 6 then in lemma 9 page 762 we do not need to require that kn 1 is a jump time provided we assume that v is nice making the additional assumption that v is nice proposition 1 and its proof and the rest of the paper go through unchanged lemma 6 for each 0 sufficiently small there exists 3 0 such that if i is a scaled neighbourhood of a nice interval v i then j is a 3 scaled neighbourhood of any component a of k j v where k 1 is arbitrary

absence of line fields and mane s theorem for nonrecurrent transcendental functions let f c c over cap be a transcendental meromorphic function suppose that the finite part p f boolean and c of the postsingular set of f is bounded that f has no recurrent critical points or wandering domains and that the degree of pre poles of f is uniformly bounded then we show that f supports no invariant line fields on its julia set we prove this by generalizing two results about rational functions to the transcendental setting a theorem of mane 1993 about the branching of iterated preimages of disks and a theorem of mcmullen 1994 regarding the absence of invariant line fields for measurably transitive functions both our theorems extend results previously obtained by graczyk kotus and swiatek 2004

combinatorial rigidity for unicritical polynomials we prove that any unicritical polynomial fc z zd c which is at most finitely renormalizable and has only repelling periodic points is combinatorially rigid this implies that the connectedness locus the multibrot set is locally connected at the corresponding parameter values and generalizes yoccoz s theorem for quadratics to the higher degree case

sampling of the wiener process for remote estimation over a channel with random delay in this paper we consider a problem of sampling a wiener process with samples forwarded to a remote estimator over a channel that is modeled as a queue the estimator reconstructs an estimate of the real time signal value from causally received samples we study the optimal online sampling strategy that minimizes the mean square estimation error subject to a sampling rate constraint we prove that the optimal sampling strategy is a threshold policy and find the optimal threshold this threshold is determined by how much the wiener process varies during the random service time and the maximum allowed sampling rate further if the sampling times are independent of the observed wiener process the above sampling problem for minimizing the estimation error is equivalent to a sampling problem for minimizing the age of information this reveals an interesting connection between the age of information and remote estimation error our comparisons show that the estimation error achieved by the optimal sampling policy can be much smaller than those of age optimal sampling zero wait sampling and periodic sampling

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

paging and registration in cellular networks jointly optimal policies and an iterative algorithm this paper explores optimization of paging and registration policies in cellular networks motion is modeled as a discrete time markov process and minimization of the discounted infinite horizon average cost is addressed the structure of jointly optimal paging and registration policies is investigated through the use of dynamic programming for partially observed markov processes it is shown that there exist policies with a certain simple form that are jointly optimal an iterative algorithm for policies with the simple form is proposed and investigated the algorithm alternates between paging policy optimization and registration policy optimization it finds a pair of individually optimal policies majorization theory and riesz s rearrangement inequality are used to show that jointly optimal paging and registration policies are given for symmetric or gaussian random walk models by the nearest location first paging policy and distance threshold registration policies

rigidity and non local connectivity of julia sets of some quadratic polynomials for an infinitely renormalizable quadratic map f c z mapsto z 2 c with the sequence of renormalization periods km and rotation numbers tm pm qm we prove that if limsup k m 1 log p m 0 then the mandelbrot set is locally connected at c we prove also that if limsup t m 1 1 q m 1 and qm then the julia set of fc is not locally connected and the mandelbrot set is locally connected at c provided that all the renormalizations are non primitive satellite this quantifies a construction of a douady and j hubbard and weakens a condition proposed by j milnor

the quasi additivity law in conformal geometry on a riemann surface s of nite type containing a family of n disjoint disks di islands we consider several natural conformal invariants measuring the distance from the islands to s and the separation between dierent islands in a near degenerate situation we establish a relation between them called the quasi additivity law we then generalize it to a quasi invariance law providing us with a transformation rule of the moduli in question under covering maps this rule and in particular its special case called the covering lemma has important applications in holomorphic dynamics

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

bounds for maps of an interval with one critical point of inflection type ii recently quite a few papers appeared proving complex bounds local connectivity of julia sets absence of invariant linefields and quasisymmetric rigidity for real polynomial maps let us first discuss the unimodal case in 1990 martens ma proved that there are real bounds for a certain sequence of first return maps to suitable central intervals in 1990 sullivan showed that if the unimodal maps are real analytic infinitely renormalizable and have bounded combinatorics then their first return maps extend to the complex plane as polynomial like maps see su and also ms in 1994 the present authors proved that all real analytic unimodal maps allow such polynomial like complex extensions for the non central first return maps the proof even gave explicit numerical lower bounds for the moduli of certain annuli ls1 for the quadratic case different proofs were later provided by gs1 ly for the smooth case see ko in 1996 sands see sa improved our numerical bounds for maps of the form g x so that the schwarzian derivative of g is negative this polynomial like structure is one of the main ingredients in many recent results on smooth unimodal maps for example it is heavily used in the proof

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

local connectivity and quasi conformal rigidity of non renormalizable polynomials we prove that topologically conjugate non renormalizable polynomials are quasi conformally conjugate from this we derive that each such polynomial can be approximated by a hyperbolic polynomial as a by product we prove that the julia set of a non renormalizable polynomial with only hyperbolic periodic points is locally connected and the branner hubbard conjecture the main tools are the enhanced nest construction developed in a previous joint paper with rigidity for real polynomials ann of math 2 165 2007 749 841 and a lemma of kahn and lyubich for which we give an elementary proof in the real case

decay of geometry for unimodal maps negative schwarzian case we show that decay of geometry holds for unimodal maps of the interval which have negative schwarzian derivative sufficient finite smoothness and a nondegenerate critical point the proof is based on pseudo analytic extensions of order at least 2 they allow us to modify sullivan s principle that rescaled high iterates of one dimensional maps tend to analytic limits in such a way that no passage to a limit is actually needed but the maps are shown to approach the analytic class in a well defined sense as a technical improvement this method yields a uniform estimate in the case of renormalizable maps

the distortion rate function of sampled wiener processes we consider the recovery of a continuous time wiener process from a quantized or a lossy compressed version of its uniform samples under limited bitrate and sampling rate we derive a closed form expression for the optimal tradeoff among sampling rate bitrate and quadratic distortion in this setting this expression is given in terms of a reverse waterfilling formula over the asymptotic spectral distribution of a sequence of finite rank operators associated with the optimal estimator of the wiener process from its samples we show that the ratio between this expression and the standard distortion rate function of the wiener process describing the optimal tradeoff between bitrate and distortion without a sampling constraint is only a function of the number of bits per sample we also consider a sub optimal lossy compression scheme in which the continuous time process is estimated from the output of an encoder that is optimal with respect to the discrete time samples we show that the latter is strictly greater than the distortion under optimal encoding but only by at most 3 we therefore conclude that near optimal performance is attained even if the encoder is unaware of the continuous time origin of the samples