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Linear Chaos?

This is an expository paper in which the author hopes to convince the reader that linear dynamics can exhibit the same beauty and complexity as non-linear dynamics.

In particular it is shown that continuous linear operators on Hilbert space can be chaotic in the sense of Devaney.  While this has been known for sometime, the author also proves a new result that shows that the orbits of linear operators can be as complicated as the orbits of any continuous function.

This paper arose from the author's line of questions asking just how complicated orbits of linear operators (that are not dense) could be.  For example, can an orbit be dense in a Cantor set?  or in a Julia set?  or other fractal-like sets?  Can the closure of an orbit for a linear operator have any prescribed Hausdorff dimension?

Surprisingly, the answer to all these questions is YES!  Furthermore, the beautiful result proven below essentially says if you can find a continuous function with an orbit having a certain property, then you can also find a linear operator with an orbit having that same property.

In what follows let H(n) be an n-dimensional separable complex Hilbert space, n = \infty is allowed.  Also, B(n) will denote the backward shift on l^2(H(n)) which is the infinite dimensional separable complex Hilbert space of all sequences in H(n) whose norms are square summable.

Theorem 1:  If f:X --> X is a continuous function on a closed bounded subset X \subseteq H(n), and T = 2B(n), then there is an invariant closed set K for T such that T|K is topologically conjugate to f.

Corollary:  There is a continuous linear operator T on a separable Hilbert space such that if f:X-->X is a continuous function on a compact metric space X, then there exists an invariant closed set K for T such that T|K is topologically conjugate to f.

That is, there is a linear operator that is universal for all continuous functions on compact metric spaces, up to topological conjugacy!

Theorem 2: If f:X --> X is a Lipschitz function on a closed bounded subset X \subseteq H(n), then there exists a c > 1 and an invariant closed set K for T = cB(n) such that T|K is topologically conjugate to f via a bi-Lipschitz homeomorphism.

Thus for example, if c is a complex number, then f(z) = z^2 + c is Lipschitz on bounded subsets of the complex plane.  Furthermore, the Julia set J for f is an invariant compact set such that f:J --> J is chaotic (has a dense orbit and a dense set of periodic points).  In particular, f has an orbit dense in J.  Thus, there are linear operators, in fact multiples of the Backward shift, that have orbits dense in sets that are bi-Lipschitz homeomorphic to Julia sets!!!