# Analytic number theory: An introduction by Richard Bellman

By Richard Bellman

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This proposal, controversial at ﬁrst, has gained much favor. It seems that Smale had very few physical motivations when cooking up his theory of hyperbolic systems, while physics itself does not seem to encompass many hyperbolic systems. This is at least Anosov’s point of view [4]: One gets the impression that the Lord God would prefer to weaken hyperbolicity a bit rather than deal with restrictions on the topology of an attractor that arise when it really is “1960s-model” hyperbolic. Even nowadays, it is not easy to ﬁnd physical phenomena with strictly hyperbolic dynamics (see however [35, 39]).

Following a nonwandering geodesic on the pants ???? , after each crossing with a seam one may consider to turn right or turn left to reach the next seam. It is thus possible to associate to each nonwandering geodesic a bi-inﬁnite sequence of “left/right” symbols. Yet, this new coding is not perfect, because it is not bijective. Assume the pants is embedded symmetrically in space, meaning that it is invariant 38 E. Ghys Figure 20. Symmetries of the pants through six rotations (the identity, two rotations of order three, and three rotations of order two), as in Figure 20.

Thus within the limits of accuracy of the printed values, the trajectory is conﬁned to a pair of surfaces which appear to merge in the lower portion. [. . ] It would seem, then, that the two surfaces merely appear to merge, and remain distinct surfaces. [. . , and we ﬁnally conclude that there is an inﬁnite complex of surfaces, each extremely close to one or the other of the two merging surfaces. Figure 12 is reprinted from Lorenz’s article. Starting from an initial condition, the orbit rapidly approaches this “two-dimensional object” and then travels “on” this surface.