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

rational rotation numbers for maps of the circle we consider families of maps of the circle of degree 1 which are homeomorphisms but not diffeomorphisms that is maps like x to x t frac c 2 pi sin 2 pi x bmod 1 withc 1 we prove that the set of parameter values corresponding to irrational rotation numbers has lebesgue measure 0 in other words the intervals on which frequency locking occurs fill up the set of full measure

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

renormalization of almost commuting pairs in this paper we give a new proof of hyperbolicity of renormalization of critical circle maps using the formalism of almost commuting pairs we extend renormalization to two dimensional dissipative maps of the annulus which are small perturbations of one dimensional critical circle maps finally we demonstrate that a two dimensional map which lies in the stable set of the renormalization operator possesses attractor which is topologically a circle such a circle is critical the dynamics on it is topologically but not smoothly conjugate to a rigid rotation

variational principles for branched circle patterns abstract we are concerned with variational principles for branched circle patterns we use critical point theory to establish existence and uniqueness furthermore we apply the ricci flow method to give an approximation scheme which converges exponentially to the branched circle pattern metric

renormalization horseshoe and rigidity for circle diffeomorphisms with breaks in this paper we build the renormalization horseshoe for the circle homeomorphisms which are c2 smooth everywhere except for one point and at that point have a jump in first derivative we also show that two such homeomorphisms are c1 smoothly conjugate for a certain class of rotation numbers which include non diophantine numbers with arbitrarily high rate of growth

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

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 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

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

a branched andreev thurston theorem for circle packings of the sphere andreev thurston theorem let r be a triangulation of s which is not simplicially equivalent to a tetrahedron and let r 0 n be given where t denotes the set of edges of x suppose that a if eu e2 e3 e r 1 form a closed loop on s and if 2 i e n then eif e2 e3 are the three edges of a two simplex of x b if ex e2 e3 ea e r 1 form a closed loop on s and if 2 i e 2k then ex e2 e2 e4 form the boundary of the union of two adjacent twosimplexes of x

quasiperiodicity in dissipative systems a renormalization group analysis abstract dynamical systems with quasiperiodic behavior i e two incommensurate frequencies may be studied via discrete maps which show smooth continuous invariant curves with irrational winding number in this paper these curves are followed using renormalization group techniques which are applied to a one dimensional system circle and also to an area contracting map of an annulus two fixed points are found representing different types of universal behavior a trivial fixed point for smooth motion and a nontrivial fixed point the latter representsthe incipient breakup of a quasiperiodic motion with frequency ratio the golden mean into a more chaotic flow fixed point functions are determined numerically and via an e expansion and eigenvalues are calculated

robust rigidity for circle diffeomorphisms with singularities we prove that under certain regularity conditions imposed on the renormalizations of two circle diffeomorphisms with singularities their c 1 smooth equivalence follows from exponential convergence of those renormalizations as an easy corollary any two analytical critical circle maps with the same order of critical points and the same irrational rotation number are c 1 smoothly conjugate

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

combinatorial ricci flows on surfaces we show that the analogue of hamilton s ricci flow in the combinatorial setting produces solutions which converge exponentially fast to thurston s circle packing on surfaces as a consequence a new proof of thurston s existence of circle packing theorem is obtained as another consequence ricci flow suggests a new algorithm to find circle packings

rigidity of infinite disk patterns let p be a locally flnite disk pattern on the complex plane c whose combinatorics are described by the one skeleton g of a triangulation of the open topological disk and whose dihedral angles are equal to a function e 0 2 on the set of edges let p be a combinatorially equivalent disk pattern on the plane with the same dihedral angle function we show that p and p difier only by a euclidean similarity in particular when the dihedral angle function is identically zero this yields the rigidity theorems of b rodin and d sullivan and of o schramm whose arguments rely essentially on the pairwise disjointness of the interiors of the disks the approach here is analytical and uses the maximum principle the concept of vertex extremal length and the recurrency of a family of electrical networks obtained by placing resistors on the edges in the contact graph of the pattern a similar rigidity property holds for locally flnite disk patterns in the hyperbolic plane where the proof follows by a simple use of the maximum principle also we have a uniformization result for disk patterns in a future paper the techniques of this paper will be extended to the case when 0 in particular we will show a rigidity property for a class of inflnite convex polyhedra in the 3 dimensional hyperbolic space

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

the hausdorff dimension of the boundary of the mandelbrot set and julia sets it is shown that the boundary of the mandelbrot set m has hausdorff dimension two and that for a generic c e am the julia set of z i z2 c also has hausdorff dimension two the proof is based on the study of the bifurcation of parabolic periodic points

the ricci flow on the 2 sphere the classical uniformization theorem interpreted differential geomet rically states that any riemannian metric on a 2 dimensional surface ispointwise conformal to a constant curvature metric thus one can con sider the question of whether there is a natural evolution equation whichconformally deforms any metric on a surface to a constant curvature met ric the primary interest in this question is not so much to give a newproof of the uniformization theorem but rather to understand nonlinearparabolic equations better especially those arising in differential geome try a sufficiently deep understanding of parabolic equations should yieldimportant new results in riemannian geometry the question in the preceding paragraph has been studied by richardhamilton 3 and brad osgood ralph phillips and peter sarnak 6 in 3 hamilton studied the following equation which we refer to as

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

variational principles for circle patterns and koebe s theorem we prove existence and uniqueness results for patterns of circles with prescribed intersection angles on constant curvature surfaces our method is based on two new functionals one for the euclidean and one for the hyperbolic case we show how colin de verdiere s bragger s and rivin s functionals can be derived from ours

on functional equations associated with the renormalization of non commuting circle mappings we investigate the functional equations associated with critical non commuting circle homeomorphisms using epstein s herglotz function methods we show that there are analytic solutions of a class of functional equations closely related to period two points of the circle map renormalization transformations for rotation number with continued fraction

branched circle packings and discrete blaschke products in this paper we introduce the notion of discrete blaschke products via circle packing we first establish necessary and sufficient conditions for the existence of finite branched circle packings next discrete blaschke products are defined as circle packing maps from univalent circle packings that properly fill d z izi 1 to the corresponding branched circle packings that properly cover d it is verified that such maps have all geometric properties of their classical counterparts finally we show that any classical finite blaschke product can be approximated uniformly on compacta of d by discrete ones

what color is your jacobian graph coloring for computing derivatives graph coloring has been employed since the 1980s to efficiently compute sparse jacobian and hessian matrices using either finite differences or automatic differentiation several coloring problems occur in this context depending on whether the matrix is a jacobian or a hessian and on the specifics of the computational techniques employed we consider eight variant vertex coloring problems here this article begins with a gentle introduction to the problem of computing a sparse jacobian followed by an overview of the historical development of the research area then we present a unifying framework for the graph models of the variant matrix estimation problems the framework is based upon the viewpoint that a partition of a matrix into structurally orthogonal groups of columns corresponds to distance 2 coloring an appropriate graph representation the unified framework helps integrate earlier work and leads to fresh insights enables the design of more efficient algorithms for many problems leads to new algorithms for others and eases the task of building graph models for new problems we report computational results on two of the coloring problems to support our claims most of the methods for these problems treat a column or a row of a matrix as an atomic entity and partition the columns or rows or both a brief review of methods that do not fit these criteria is provided we also discuss results in discrete mathematics and theoretical computer science that intersect with the topics considered here

on the existence of fixed points of the composition operator for circle maps in the theory of circle maps with golden ratio rotation number formulated by feigenbaum kadanoff and shenker fks and by ostlund rand sethna and siggia orss a central role is played by fixed points of a certain composition operator in map space we define a common setting for the problem of proving the existence of these fixed points and of those occurring in the theory of maps of the interval we give a proof of the existence of the fixed points for a wide range of the parameters on which they depend