If you stare at fractals long enough, you start seeing things. Consider, for example, the Mandelbrot Set, or DLA (Diffusion-Limited Aggregation). There is a natural potential you can assign to these things, the Douady-Hubbard potential. The "rays" of "field lines" of this potential are geodesics. These geodesics start far away from the M-set, and as you follow them in, they get twistier and turnier as they approach the M-Set. The Douady-Hubbard potential is extremely hyperbolic.

If you were a flat-lander, and these rays were your life-geodesic, that is, if the movement along these rays constituted the passage of time, you'd notice a certain kind of variety and monotony in your life. As your life passed by one feature of the M-set, before long, you'd come on another, which, it would look like the first, but slightly different. After all, this is what "self similarity" for a fractal is. If you were a flatlander, following your geodesic, day after day, your life might sound like this: "Today I went to the office, left for lunch, and then came back to the office, which looked exactly like the office that I'd left earlier. Tonight, I'm going home, and tomorrow, it will be another day at the office, which will look almost exactly like the office today does, but of course will be slightly different, because it occurs later along my timeline geodesic". I think Plato expressed this along the lines of "you can never step into the same river twice". If your life was described by a geodesic on a DLA, or the M-Set, that would be true. But self-similarity means that tomorrow would look a lot like today.

If you were a flat-lander living on a geodesic, would you have free will? The space in which you live is very hyperbolic, and all geodesics have a positive Lyapunov exponent. It would take an infinitesimal nudge to move you from one life to a completely different one. How hyperbolic is it? Well, the modular group SL(2,Z) plays a role in describing the self similarity of the M-Set. This hyperbolic space seems to be isomorphic to the upper-half-plane, with the Poincare metric that is invariant under SL(2,R). So this funny flatland does resemble the higher-dimensional Minkowski space, with its SL(2,C) invariance in a certain way. Except that we get we get three spatial dimensions to our time-like geodesic, thanks to the spinor structure of SL(2,C), whereas the flatlander, following their little Douady-Hubbard, SL(2,R) time-like geodesic, gets only one spatial dimension. But life near the M-Set can still be filled with surprises for flatlanders, I suppose.

The strange thing for us is, we live in this hyperbolic Minkowski space, but we don't seem to much notice positive Lyapunov exponents in our lives. Other than that nature all around us seems to be fractal and self-similar ... other than that, we don't notice. I sometimes wonder if the quantum-entanglement problems of quantum-mechanics are really some manifestation of life in Minkowski space, some deep connection we haven't noticed just yet. Some deep statement about Hamiltonians on hyperbolic manifolds that we don't yet know, some fundamental theorem of analysis and algebra that we haven't quite yet uncovered. For example: We think that the Riemann zeros might be the eigenvalues of some very simple quantum Hamiltonian (this is Berry's statement). And we know that the Riemann zeta in intimately tied into SL(2,R) and hyperbolic dynamics in general. (I have web pages on this). So is general quantum entanglement some statement about Hamiltonians in hyperbolic spaces? It is very tempting to think so ...

You know what else is highly hyperbolic? Free groups. Take a look at the Cayley graph of the free group generated by {a,b}. Its very fractal, its very self-similar, its very hyperbolic. Any tree graph is essentially hyperbolic: as you follow a tree, say a binary tree, at every branch, every fork in the road, you have a choice, "shall I go left, or shall I go right"? Paths on trees are necessarily divergent, necessarily have a positive Lyapunov exponent. Binary trees are explicitly homomorphic to the modular group SL(2,R); they rotations and rebalancings of binary trees correspond to group elements.

You know where else the free group generated by {a,b} plays an important role? Its central to the proof of the Banach-Tarski paradox. The Banach-Tarski paradox is essentially the observation that there are these immense subsets of three-dimensional space that are not measurable, in the sense of "Hausdorff measure" or "Riemann-Stieltjes integral", or any of the "measures" commonly employed by mathematicians to denote the size of a set. This result is not limited to 3D space, there are variants of the Banach-Tarski paradox for any dimension. Physicists, when they solve their differential equations, or when they evolve their operators in Hilbert space, like to think that the dynamical systems that they are solving are somehow defined for "all" points in the set of "real numbers". Its quite beguiling: a smooth, continuous, differentiable function of a real variable x is definable for "any" real value. And so in quantum mechanics (and in fluid mechanics) one talks about functions defined for "all" space. But the space of functions is not really like the topological space of plain-old "real numbers". The Banach-Tarski paradox reminds us that infinitesimally close to any real number is a giant, immense set of other real numbers that is not measurable in the classical sense, cannot be made to take part in, or contribute to, integrals and derivatives in the classical, analytic sense. So when a physicist, whether it be Newton or anyone since who subscribes to the philosophical belief in determinism, who insists on the idea that dynamical equations and geodesics prove that we have no free will and cannot make choices about outcomes, watch out. It is only because they are ignorant of Banach-Tarski. The space of real points, the topological space of manifolds (and the particular Minkowski space we live in) is not isomorphic to the space of solutions to some quantum Hamiltonian. Minkowski space is not isomorphic to the space of solutions of the "Quantum wave function of the Universe". Such a wave function does arguably exist, and it does arguably describe matter in the universe. But, and it is a big "but", this theory of operators and wave functions has problems with the fact that you can slice and dice manifolds with Banach-Tarski and a pinch of "Axiom of Choice". (Dang, I thought I could finish that paragraph without using those words. Oh well).

The above paragraphs are *not* purely speculative. There are honest-to-goodness models one can study. Let us take the unit interval [0,1] of the real number line, and partition it into two Banach-Tarski sets. Dynamical systems in two dimensions have geodesics that can be labelled with points on the real-number line. When the dynamical system is hyperbolic, these geodesics have a positive Lyapunov exponent: they diverge. Two infinitesimally close geodesics eventually diverge. How does one generate Banach-Tarski type sets? Take a pair of incommensurate irrationals, and iterate on them (i.e. call them theta, and consider the set exp i n theta): create their free group. Create the cosets. Which geodesics belong to which cosets? One the one hand, the geodesics, by construction, cannot be identical, as they are intimately tied to the irrational that generated the cosets. And yet, by axiom of choice, they really are "the same thing". OK, I admit, I made a hash of the above paragraph. But the point is that we have solutions to dynamical systems, i.e. geodesics, that can be partitioned into *disjoint* sets, and that these disjoint sets on the one hand are identical, and on the other hand describe the trajectories of neighboring points. Hmm. I dunno. Does this idea have legs? Maybe not. If I only wait a finite amount of time, then two trajectories in these disjoint sets have not yet diverged. Hmmm...

Copyrighted under the Gnu FDL. Last modified by Linas Vepstas on 15 January 2005.