Posted by: **Dave Richeson** | September 6, 2009

## Symbolic dynamics for nonhyperbolic systems

I suppose the intersection of people who read this blog and people who are interested in my research is, if not zero, close to zero (come to think of it, the set of people interested in my research is not much greater than zero as it is). But I thought I’d mention that my collaborator (Jim Wiseman) and I posted a paper to the arXiv on Friday and then sent it to a journal. The paper is called “Symbolic dynamics for nonhyperbolic systems.” Here’s the abstract, feel free to check it out.

We introduce index systems, a tool for studying isolated invariant sets of dynamical systems that are not necessarily hyperbolic. The mapping of the index systems mimics the expansion and contraction of hyperbolic maps on the tangent space, and they may be used like Markov partitions to generate symbolic dynamics. Every continuous dynamical system satisfying a weak form of expansiveness possesses an index system. Because of their topological robustness, they can be used to obtain rigorous results from computer approximations of a dynamical system.

I was working on two papers over the summer, hoping to finish them both before classes started. I didn’t meet that goal, but at least this one was sent off before the end of the first week of classes. The second one is mostly done, but now that the students have returned and my role as department chair has kicked back into high gear, I’m not sure when it will get finished.

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