Chaotic Period Doubling
01 Apr 2009-Ergodic Theory and Dynamical Systems (Cambridge University Press)-Vol. 29, Iss: 2, pp 381-418
TL;DR: In this article, it was shown that the period doubling renormalization operator acting on the space of C1+Lip unimodal maps has infinite topological entropy and that the analytic fixed point is not hyperbolic.
Abstract: The period doubling renormalization operator was introduced by Feigenbaum and by Coullet and Tresser in the 1970s to study the asymptotic small-scale geometry of the attractor of one-dimensional systems that are at the transition from simple to chaotic dynamics. This geometry turns out not to depend on the choice of the map under rather mild smoothness conditions. The existence of a unique renormalization fixed point that is also hyperbolic among generic smooth-enough maps plays a crucial role in the corresponding renormalization theory. The uniqueness and hyperbolicity of the renormalization fixed point were first shown in the holomorphic context, by means that generalize to other renormalization operators. It was then proved that, in the space of C2+alpha unimodal maps, for alpha > 0, the period doubling renormalization fixed point is hyperbolic as well. In this paper we study what happens when one approaches from below the minimal smoothness thresholds for the uniqueness and for the hyperbolicity of the period doubling renormalization generic fixed point. Indeed, our main result states that in the space of C2+alpha unimodal maps the analytic fixed point is not hyperbolic and that the same remains true when adding enough smoothness to get a priori bounds. In this smoother class, called C2+vertical bar center dot vertical bar, the failure of hyperbolicity is tamer than in C-2. Things get much worse with just a bit less smoothness than C-2, as then even the uniqueness is lost and other asymptotic behavior becomes possible. We show that the period doubling renormalization operator acting on the space of C1+Lip unimodal maps has infinite topological entropy.
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TL;DR: In this paper, the authors present several approaches to renormalization in QFT: the multi-scale analysis in perturbative renormalisation, the functional methods, and the loop-vertex expansion.
Abstract: We present several approaches to renormalization in QFT: the multi-scale analysis in perturbative renormalization, the functional methods \`a la Wetterich equation, and the loop-vertex expansion in non-perturbative renormalization. While each of these is quite well-established, they go beyond standard QFT textbook material, and may be little-known to specialists of each other approach. This review is aimed at bridging this gap.
36 citations
TL;DR: For the shift σ in Σ = {0, 1}ℕ, this article defined the renormalization for potentials by defining a unique fixed point for a good H, such that σ 2 σ H = H ◦ σ.
Abstract: For the shift σ in Σ = {0, 1}ℕ, we define the renormalization for potentials by We show that for a good H, there is a unique fixed point for . It is the Hofbauer potential V*. We show that the stable set of the Hofbauer potential, i. e. the set of potentials V such that converges to V* is characterized by the germ of these potentials close to 0∞ = 000…. Then, we make connections with the Manneville–Pomeau map f : [0, 1]↺. In particular we show that the lift in Σ of log f′ is in the stable set of V*. In the second part, we characterize "good" H, such that σ2 ◦ H = H ◦ σ. In the last part, we study the thermodynamic formalism for some special potentials in the stable set of V*. They are called virtual Manneville–Pomeau maps.
27 citations
01 Jan 2011
TL;DR: In this article, the authors considered strongly dissipative Henon maps as perturbations of one-dimensional systems and showed that the properties of the Henon attractors do not depend on the actual system, they are universal.
Abstract: Period doubling cascades are observed at transition to chaos in many models used in the sciences and in physical experiments. These period doubling cascades are very well understood in one-dimensional dynamics. In particular, the microscopic geometrical properties of the attractors do not depend on the actual system, they are universal. Moreover, the attractors of two different maps are smoothly conjugate, they are rigid. Strongly dissipative Henon maps describe parts of the dynamics of systems close to a homoclinic tangency and are often observed in various models. For these maps the transition to positive entropy also occurs along period doubling cascades. These strongly dissipative Henon maps can be considered as perturbations of one-dimensional systems. Indeed, some of the universal geometrical properties of the one-dimensional systems are present in the Henon maps. However, they appear in a much more delicate form: in a probabilistic sense the geometry of the Henon attractors is the same as their one-dimensional counter part. This phenomenon is revered to as probabilistic universality and rigidity.
13 citations
TL;DR: In this article, the authors construct examples of infinitely renormalizable unimodal Lorenz maps which do not have a physical measure on the attractor of a system and show that the position of the critical point of the consecutive renormalizations is a crucial technical ingredient used to obtain these examples.
Abstract: A physical measure on the attractor of a system describes the statistical behavior of typical orbits. An example occurs in unimodal dynamics: namely, all infinitely renormalizable unimodal maps have a physical measure. For Lorenz dynamics, even in the simple case of infinitely renormalizable systems, the existence of physical measures is more delicate. In this article, we construct examples of infinitely renormalizable Lorenz maps which do not have a physical measure. A priori bounds on the geometry play a crucial role in (unimodal) dynamics. There are infinitely renormalizable Lorenz maps which do not have a priori bounds. This phenomenon is related to the position of the critical point of the consecutive renormalizations. The crucial technical ingredient used to obtain these examples without a physical measure is the control of the position of these critical points.
10 citations
TL;DR: In this article, the authors describe the theory of renormalization for Lorenz maps with a critical point and prove that a restriction of the renormalisation operator acting on such maps has a hyperbolic fixed point.
Abstract: A Lorenz map is a Poincare map for a three-dimensional Lorenz flow. We describe the theory of renormalization for Lorenz maps with a critical point and prove that a restriction of the renormalization operator acting on such maps has a hyperbolic fixed point. The proof is computer assisted and we include a detailed exposition on how to make rigorous estimates using a computer as well as the implementation of the estimates.
10 citations
References
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TL;DR: In this article, a large class of recursion relations xn+l = Af(xn) exhibiting infinite bifurcation is shown to possess a rich quantitative structure essentially independent of the recursion function.
Abstract: A large class of recursion relations xn+l = Af(xn) exhibiting infinite bifurcation is shown to possess a rich quantitative structure essentially independent of the recursion function. The functions considered all have a unique differentiable maximum 2. With f(2) - f(x) ~ Ix - 21" (for Ix - 21 sufficiently small), z > 1, the universal details depend only upon z. In particular, the local structure of high-order stability sets is shown to approach universality, rescaling in successive bifurcations, asymptotically by the ratio c~ (a = 2.5029078750957... for z = 2). This structure is determined by a universal function g*(x), where the 2"th iterate off, f("~, converges locally to ~-"g*(~nx) for large n. For ithe class of f's considered, there exists a A~ such that a 2"-point stable limit cycle including :7 exists; A~ - ~ ~ ~-" (~ = 4.669201609103... for z = 2). The numbers = and have been computationally determined for a range of z through their definitions, for a variety off's for each z. We present a recursive mechanism that explains these results by determining g* as the fixed-point (function) of a transformation on the class off's. At present our treatment is heuristic. In a sequel, an exact theory is formulated and specific problems of rigor isolated.
2,984 citations
Book•
01 May 1976
TL;DR: Ma as mentioned in this paper introduces the beginner to fundamental theoretical concepts such as mean field theory, scaling hypothesis, and renormalization group, with emphasis on the underlying physics and the basic assumptions involved.
Abstract: An important contributor to our current understanding of critical phenomena, Ma introduces the beginner--especially the graduate student with no previous knowledge of the subject-to fundamental theoretical concepts such as mean field theory, the scaling hypothesis, and the renormalization group. He then goes on to apply the renormalization group to selected problems, with emphasis on the underlying physics and the basic assumptions involved.
2,300 citations
TL;DR: In this paper, the role of functional equations to describe the exact local structure of highly bifurcated attractors is formally developed, and a hierarchy of universal functions, each descriptive of the same local structure but at levels of a cluster of 2>>\s points, is presented.
Abstract: The role of functional equations to describe the exact local structure of highly bifurcated attractors ofx
n+1
=λf(x
n
) independent of a specificf is formally developed. A hierarchy of universal functionsg
r
(x) exists, each descriptive of the same local structure but at levels of a cluster of 2
r
points. The hierarchy obeysg
r−1
(x)=−αg
r(gr(x/α), withg=limr → ∞ gr existing and obeyingg(x) = −αg(g(x/α), an equation whose solution determines bothg andα. Forr asymptoticg
r
∼ g − δ−r
h
* where δ > 1 andh are determined as the associated eigenvalue and eigenvector of the operator ℒ:
$$\mathcal{L}\left[ \psi \right] = - \alpha \left[ {\psi \left( {g\left( {{x \mathord{\left/ {\vphantom {x \alpha }} \right. \kern-
ulldelimiterspace} \alpha }} \right)} \right) + g'\left( {g\left( {{x \mathord{\left/ {\vphantom {x \alpha }} \right. \kern-
ulldelimiterspace} \alpha }} \right)} \right)\psi \left( {{{ - x} \mathord{\left/ {\vphantom {{ - x} \alpha }} \right. \kern-
ulldelimiterspace} \alpha }} \right)} \right]$$
We conjecture that ℒ possesses a unique eigenvalue in excess of 1, and show that this δ is the λ-convergence rate. The form (*) is then continued to allλ rather than just discreteλ
r
and bifurcation valuesΛ
r
and dynamics at suchλ is determined. These results hold for the high bifurcations of any fundamental cycle. We proceed to analyze the approach to the asymptotic regime and show, granted ℒ's spectral conjecture, the stability of theg
r
limit of highly iterated λf's, thus establishing our theory in a local sense. We show in the course of this that highly iterated λf's are conjugate tog
r
's, thereby providing some elementary approximation schemes for obtainingλ
r
for a chosenf.
1,160 citations
Book•
01 Jan 1993
TL;DR: In this article, the authors propose a topological theory of circle Diffeomorphisms and the Combinatorics of one-dimensional endomorphisms, based on the theory of Denjoy inequalities.
Abstract: 0. Introduction.- I. Circle Diffeomorphisms.- 1. The Combinatorial Theory of Poincare.- 2. The Topological Theory of Denjoy.- 2.a The Denjoy Inequality.- 2.b Ergodicity.- 3. Smooth Conjugacy Results.- 4. Families of Circle Diffeomorphisms Arnol'd tongues.- 5. Counter-Examples to Smooth Linearizability.- 6. Frequency of Smooth Linearizability in Families.- 7. Some Historical Comments and Further Remarks.- II. The Combinatorics of One-Dimensional Endomorphisms.- 1. The Theorem of Sarkovskii.- 2. Covering Maps of the Circle as Dynamical Systems.- 3. The Kneading Theory and Combinatorial Equivalence.- 3.a Examples.- 3.b Hofbauer's Tower Construction.- 4. Full Families and Realization of Maps.- 5. Families of Maps and Renormalization.- 6. Piecewise Monotone Maps can be Modelled by Polynomial Maps.- 7. The Topological Entropy.- 8. The Piecewise Linear Model.- 9. Continuity of the Topological Entropy.- 10. Monotonicity of the Kneading Invariant for the Quadratic Family.- 11. Some Historical Comments and Further Remarks.- III. Structural Stability and Hyperbolicity.- 1. The Dynamics of Rational Mappings.- 2. Structural Stability and Hyperbolicity.- 3. Hyperbolicity in Maps with Negative Schwarzian Derivative.- 4. The Structure of the Non-Wandering Set.- 5. Hyperbolicity in Smooth Maps.- 6. Misiurewicz Maps are Almost Hyperbolic.- 7. Some Further Remarks and Open Questions.- IV. The Structure of Smooth Maps.- 1. The Cross-Ratio: the Minimum and Koebe Principle.- l.a Some Facts about the Schwarzian Derivative.- 2. Distortion of Cross-Ratios.- 2.a The Zygmund Conditions.- 3. Koebe Principles on Iterates.- 4. Some Simplifications and the Induction Assumption.- 5. The Pullback of Space: the Koebe/Contraction Principle.- 6. Disjointness of Orbits of Intervals.- 7. Wandering Intervals Accumulate on Turning Points.- 8. Topological Properties of a Unimodal Pullback.- 9. The Non-Existence of Wandering Intervals.- 10. Finiteness of Attractors.- 11. Some Further Remarks and Open Questions.- V. Ergodic Properties and Invariant Measures.- 1. Ergodicity, Attractors and Bowen-Ruelle-Sinai Measures.- 2. Invariant Measures for Markov Maps.- 3. Constructing Invariant Measures by Inducing.- 4. Constructing Invariant Measures by Pulling Back.- 5. Transitive Maps Without Finite Continuous Measures.- 6. Frequency of Maps with Positive Liapounov Exponents in Families and Jakobson's Theorem.- 7. Some Further Remarks and Open Questions.- VI. Renormalization.- 1. The Renormalization Operator.- 2. The Real Bounds.- 3. Bounded Geometry.- 4. The PullBack Argument.- 5. The Complex Bounds.- 6. Riemann Surface Laminations.- 7. The Almost Geodesic Principle.- 8. Renormalization is Contracting.- 9. Universality of the Attracting Cantor Set.- 10. Some Further Remarks and Open Questions.- VII. Appendix.- 1. Some Terminology in Dynamical Systems.- 2. Some Background in Topology.- 3. Some Results from Analysis and Measure Theory.- 4. Some Results from Ergodic Theory.- 5. Some Background in Complex Analysis.- 6. Some Results from Functional Analysis.
1,048 citations
946 citations