The differentiability of the conjugation of certain diffeomorphisms of the circle

COMPLEX DYNAMICS AND RENORMALIZATION (Annals of Mathematics Studies 135) By Curtis T. McMullen: 214 pp., £35.00, ISBN 0 691 02981 4 (Princeton University Press, 1995).

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.

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Renormalization: an advanced overview

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.

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The potential point of view for renormalization

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.

Renormalization of Hénon Maps

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.

Chaotic Period Doubling

The period doubling renormalization operator was introduced by M. Feigenbaum and by P. Coullet and C. Tresser in the nineteen-seventieth to study the asymptotic small scale geometry of the attractor of one-dimensional systems which are at the transition from simple to chaotic dynamics. This geometry turns out to not depend on the choice of the map under rather mild smoothness conditions. The existence of a unique renormalization fixed point which 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 $C^{2+\alpha}$ unimodal maps, for $\alpha$ close to one, 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 results states that in the space of $C^2$ 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 $C^{2+|\cdot|}$ the failure of hyperbolicity is tamer than in $C^2$. Things get much worse with just a bit less of smoothness than $C^2$ as then even the uniqueness is lost and other asymptotic behavior become possible. We show that the period doubling renormalization operator acting on the space of $C^{1+Lip}$ unimodal maps has infinite topological entropy.

V. V. M. S. Chandramouli

Papers

Chaotic Period Doubling

Chaotic Period Doubling