Conformal measures and Hausdorff dimension for infinitely renormalizable quadratic polynomials

We show in this paper that iff is a quadratic infinitely many times renormalizable polynomial of sufficient high combinatorial type, then: HD (J(f))= inf{: -conformal measure for f} We use Lyubich's construction of the principal nest ([Lyu97]) in order to prove this result.Partially supported by CNP q-Brazil grant # 300534/96-5     

Multipliers of periodic orbits of quadratic polynomials and the parameter plane

We prove a result about an extension of the multiplier of an attracting periodic orbit of a quadratic map as a function of the parameter. This has applications to the problem of geometry of the Mandelbrot and Julia sets. In particular, we prove that the size of p/q-limb of a hyperbolic component of the Mandelbrot set of period n is O(4 ⁿ /p), and give an explicit condition on internal arguments under which the Julia set of corresponding (unique) infinitely renormalizable quadratic polynomial is not locally connected. In memory of my grandmother Esfir Garbuz     

Infinitely renormalizable quadratic polynomials, with non-locally connected Julia set

We present two strategies for producing and describing some connected non-locally connected Julia sets of infinitely renormalizable quadratic polynomials. By using a more general strategy, we prove that all of these Julia sets fail to be arc-wise connected, and that their critical point is non-accessible. Using the first strategy we prove the existence of polynomials having an explicitly given external ray accumulating two particular, symmetric points. The limit Julia set resembles in a certain way the classical non-locally connected set: “the topologists spiral.”     

On some questions related to the Gauss conjecture for function fields

We show that, for any finite field Fq, there exist infinitely many real quadratic function fields over Fq such that the numerator of their zeta function is a separable polynomial. As pointed out by Anglès, this is a necessary condition for the existence, for any finite field Fq, of infinitely many real function fields over Fq with ideal class number one (the so-called Gauss conjecture for function fields). We also show conditionally the existence of infinitely many real quadratic function fields over Fq such that the numerator of their zeta function is an irreducible polynomial.     

Logarithmic Capacity and Renormalizability for Landing on the Mandelbrot Set

We study renormalizability of external angles of the Mandelbrotset M. Estimates are made of the logarithmic capacity of setsof angles that are infinitely renormalizable with a specificsequence of periods, using a substitution due to Douady. Theseshow that many of the infinitely renormalizable rays do landon M, which provides further evidence in support of the conjecturethat M is locally connected.     

Eigenvaluations

We study the dynamics in C² of superattracting fixed point germs and of polynomial maps near infinity. In both cases we show that the asymptotic attraction rate is a quadratic integer, and construct a plurisubharmonic function with the adequate invariance property. This is done by finding an infinitely near point at which the map becomes rigid: the critical set is contained in a totally invariant set with normal crossings. We locate this infinitely near point through the induced action of the dynamics on a space of valuations. This space carries an R-tree structure and conveniently encodes local data: an infinitely near point corresponds to an open subset of the tree. The action respects the tree structure and admits a fixed point—or eigenvaluation—which is attracting in a certain sense. A suitable basin of attraction corresponds to the desired infinitely near point.     

Infinitely renormalizable diffeomorphisms of the disk at the boundary of chaos

We show that, among area contracting embeddings of the 2-disk, infinitely renormalizable maps with a bounded geometry either have positive topological entropy or correspond to a cascade of period doubling.

     

Realizing Alexander polynomials by hyperbolic links

We realize a given (monic) Alexander polynomial by a (fibered) hyperbolic arborescent knot and link having any number of components, and by infinitely many such links having at least 4 components. As a consequence, a Mahler measure minimizing polynomial, if it exists, is realized as the Alexander polynomial of a fibered hyperbolic link of at least 2 components. For a given polynomial, we also give an upper bound for the minimal hyperbolic volume of knots/links realizing the polynomial and, in the opposite direction, construct knots of arbitrarily large volume, which are arborescent, or have given free genus at least 2.     

Phase space universality for multimodal maps

We study the dynamics of the renormalization operator for multimodal maps. In particular, we develop a combinatorial theory for certain kind of multimodal maps. We also prove that renormalizations of infinitely renormalizable multimodal maps with same bounded combinatorial type are exponentially close. Our results imply, for instance, the existence and uniqueness of periodic points for the renormalization operator with arbitrary combinatorial type.     

Attractors and recurrence for dendrite-critical polynomials

We call a rational map f dendrite-critical if all its recurrent critical points either belong to an invariant dendrite D or have minimal limit sets. We prove that if f is a dendrite-critical polynomial, then for any conformal measure μ either for almost every point its limit set coincides with the Julia set of f, or for almost every point its limit set coincides with the limit set of a critical point c of f. Moreover, if μ is non-atomic, then c can be chosen to be recurrent. A corollary is that for a dendrite-critical polynomial and a non-atomic conformal measure the limit set of almost every point contains a critical point.     

Quasisymmetric robustness of the Collet-Eckmann condition in the quadratic family

We consider quasisymmetric reparametrizations of the parameter space of the quadratic family. We prove that the set of quadratic maps which are either regular or Collet-Eckmann with polynomial recurrence of the critical orbit has full Lebesgue measure, for any such reparametrization.     

Topological lower bounds on the distance between area preserving diffeomorphisms

Area preserving diffeomorphisms of the 2-disk which are Identity near the boundary form a group which can be equipped, using theL ²-norm on its Lie algebra, with a right invariant metric. In this paper we give a lower bound on the distance between diffeomorphisms which is invariant under area preserving changes of coordinates and which improves the lower bound induced by the Calabi invariant. In the case of renormalizable and infinitely renormalizable maps, our estimate can be improved and computed.     

Renormalization in the Hénon family,II: the heteroclinic web

We study highly dissipative Hénon maps

$F_{c,b}: (x,y) \mapsto (c-x^2-by, x)$

with zero entropy. They form a region Π in the parameter plane bounded on the left by the curve W of infinitely renormalizable maps. We prove that Morse-Smale maps are dense in Π, but there exist infinitely many different topological types of such maps (even away from W). We also prove that in the infinitely renormalizable case, the average Jacobian b _F on the attracting Cantor set \({\mathcal{O}}_{F}\) is a topological invariant. These results come from the analysis of the heteroclinic web of the saddle periodic points based on the renormalization theory. Along these lines, we show that the unstable manifolds of the periodic points form a lamination outside \({\mathcal{O}}_{F}\) if and only if there are no heteroclinic tangencies.

     

Optimally accurate Petrov-Galerkin method of finite elements

We perform analysis for a finite elements method applied to the singular self-adjoint problem. This method uses continuous piecewise polynomial spaces for the trial and the test spaces. We fit the trial polynomial space by piecewise exponentials and we apply so exponentially fitted Galerkin method to singular self-adjoint problem by approximating driving terms by Lagrange piecewise polynomials, linear, quadratic and cubic. We measure the erroe in max norm. We show that method is optimal of the first order in the error estimate. We also give numerical results for the Galerkin approximation.     

The complexity of approximating a nonlinear program

We consider the problem of finding the maximum of a multivariate polynomial inside a convex polytope. We show that there is no polynomial time approximation algorithm for this problem, even one with a very poor guarantee, unless P = NP. We show that even when the polynomial is quadratic (i.e. quadratic programming) there is no polynomial time approximation unless NP is contained in quasi-polynomial time.Our results rely on recent advances in the theory of interactive proof systems. They exemplify an interesting interplay of discrete and continuous mathematics—using a combinatorial argument to get a hardness result for a continuous optimization problem.     

On the Symmetric H q -Semiclassical Polynomial Sequences of Even Class. Some Examples from the Class Two

In the present work, we deal with the quadratic decomposition of symmetric H _q -semiclassical polynomial sequences of even class. Some examples from class two are settled. We give an integral and discrete measure representations for each of considered symmetric forms.     

The full renormalization horseshoe for unimodal maps of higher degree: exponential contraction along hybrid classes

We prove exponential contraction of renormalization along hybrid classes of infinitely renormalizable unimodal maps (with arbitrary combinatorics), in any even degree d. We then conclude that orbits of renormalization are asymptotic to the full renormalization horseshoe, which we construct. Our argument for exponential contraction is based on a precompactness property of the renormalization operator (“beau bounds”), which is leveraged in the abstract analysis of holomorphic iteration. Besides greater generality, it yields a unified approach to all combinatorics and degrees: there is no need to account for the varied geometric details of the dynamics, which were the typical source of contraction in previous restricted proofs.     

Spectral Triple and Sinai–Ruelle–Bowen Measures

In this article, we recover the Sinai–Ruelle–Bowen measure associated to a real-valued Hölder continuous function defined on the Julia set of a hyperbolic quadratic polynomial, as a noncommutative measure by constructing an appropriate spectral triple.     

Measures of pseudorandomness for binary sequences constructed using finite fields

We extend the results of Goubin, Mauduit and Sárközy on the well-distribution measure and the correlation measure of order k of the sequence of Legendre sequences with polynomial argument in several ways. We analyze sequences of quadratic characters of finite fields of prime power order and consider in each case two, in general, different definitions of well-distribution measure and correlation measure of order k, respectively.     

Error Bounds for the Solution Sets of Quadratic Complementarity Problems

In this article, two types of fractional local error bounds for quadratic complementarity problems are established, one is based on the natural residual function and the other on the standard violation measure of the polynomial equalities and inequalities. These fractional local error bounds are given with explicit exponents. A fractional local error bound with an explicit exponent via the natural residual function is new in the tensor/polynomial complementarity problems literature. The other fractional local error bounds take into account the sparsity structures, from both the algebraic and the geometric perspectives, of the third-order tensor in a quadratic complementarity problem. They also have explicit exponents, which improve the literature significantly.