Detection of renewal system factors via the Conley index

Abstract:

Let $N$ be an isolating neighborhood for a map $f$. If we can decompose $N$ into the disjoint union of compact sets $N_1$ and $N_2$, then we can relate the dynamics on the maximal invariant set $\operatorname {Inv} N$ to the shift on two symbols by noting which component of $N$ each iterate of a point $x\in \operatorname {Inv} N$ lies in. We examine a method, based on work by Mischaikow, Szymczak, et al., for using the discrete Conley index to detect explicit subshifts of the shift associated to $N$. In essence, we measure the difference between the Conley index of $\operatorname {Inv}N$ and the sum of the indices of $\operatorname {Inv} N_1$ and $\operatorname {Inv} N_2$.