The Lorenz Renormalization Conjecture

Abstract

Universality in low-dimensional dynamical systems is the remarkable phenomenon that the geometry of a system at a phase transition, as well as the bifurcation patterns of nearby systems, is determined by its topology. This can be explained by an associated renormalization operator being a contraction on topological conjugacy classes. For more physically realistic systems this paradigm is too simple. In the context of Lorenz dynamics, which is associated with homoclinic bifurcations of flows, the renormalization behavior is much richer. Depending on topology, one finds either the traditional universality or two more intricate classes of dynamics. A conjecture describing the dynamics of the Lorenz renormalization operator in terms of these three classes of behavior is stated and the consequences each case has on the dynamics of Lorenz maps is discussed. Numerical evidence supporting the conjecture is provided.

Keywords:

• renormalization
• rigidity
• universality

AMS Subject Classification:

• 37E20
• 37E05

Notes

1 For a more complete list of references, see e.g. [Martens and Winckler 17].

2 Recall that the first-return map g to J (induced by f) is defined as follows: let $J\prime$ consist of all $x\in J$ for which $\exists k\ge 1$ such that f is defined at $f^j(x),\forall j=0,\dots ,k-1,$ and such that $f^k(x)\in J;$ given $x\in J\prime$ let the first-return time $\tau (x)$ be the smallest $k\ge 1$ such that $f^k(x)\in J;$ then $g:J\prime \subset J\to J$ is defined by $g(x)=f^{\tau (x)}(x).$

3 For unimodal f_k we define ${\pi }_k(i)=j$ iff $f_k^j(J_{k+1})$ is the i-th interval as they are embedded in the real line counting from the critical value; i.e. ${\pi }_k(1)=1,{\pi }_k(2)=i$ where $f_k^i(J_{k+1})$ is the interval adjacent to $f_k(J_{k+1}),{\pi }_k(3)=j$ where $f_k^j(J_{k+1})$ is the interval adjacent to $f_k^i(J_{k+1})$ which is not $f(J_{k+1}),$ and so on.

4 This should hold independently of the order of the critical point, but has so far only been proved under the assumption that the order is an integer (even for unimodal maps, odd for circle maps).

5 Even though f is undefined at c, its branches continuously extend to c since f is bounded.

6 In the notation $[x,c)$ for a half-open interval x > c is allowed.

7 In particular, if f is both w–renormalizable and $w\prime$–renormalizable (and $w\ne w\prime$), then $w_0^{\prime }=w_0{w}_1\cdots$ and $w_1^{\prime }=w_1{w}_0\cdots ,$ or vice versa, see [Martens and de Melo 01].

8 Just as the one unstable direction for unimodal renormalization is related to moving the one critical value.

9 That is, the critical point is $c(f)=0.5$ and $1-f(x)=f(1-x).$

10 The name comes from the fact that in the end x_k will be actual orbits of the critical values 0 and 1 under some map f in the family; i.e. $x_k(j)=f^j(k).$

11 For example, once (2, 1)–renormalizable type is given by (011, 10) and twice (2, 1)–renormalizable is given by (0111010, 10011).

Additional information

Funding

This project has received funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No 743959.