DMS Topology Seminar

Speaker: Stu Baldwin (Auburn)

Title: Itinerary Topologies and Wildcard Topologies in Symbolic Dynamics 

Abstract: Let \(X\) be a topological space, \(f:X \to X\) continuous, and let \(P=\{A_k:K \in K\}\) be a partition of \(X\) into \(k\) nonempty disjoint subsets whose union is \(X\) (where \(K\) is some index set having \(k\) elements).  If \(x \in X\), then the itinerary of \(x\) with respect to \(f\) and \(P\) is the sequence \(\langle i_n(x):n\in\omega \rangle\) given by \(i_n(x)=k\) iff \(f^n(x) \in A_k\) (where \(\omega\) is the set of nonnegative integers).  This gives an `itinerary function' \(i:X \to P^\omega\) which has been used for many years to study the dynamics of \(f\) (i.e., the long term behavior of the functions \(f^n\) as \(n\) gets large).  The idea works best if the sets \(A_k\) are all closed, which means that various compromises have to be made to use the idea on connected sets.  If \(I\) is an interval and \(f:I \to I\) has a `turning point' \(c\) in the interior of \(I\) (i.e., \(f\) is increasing for \(x<c\) and decreasing for \(x>c\), or vice versa), one approach is to let \(L=\{x:x<c\}\), \(M=\{c\}\), and \(R=\{x:x>c\}\), and use sequences of the three sets \(L,M,R\) (viewed as symbols) to denote the itineraries of points, and deal with the complications caused by \(L\) and \(R\) not being closed as they arise.  Other approaches have included using \(L'=L \cup M\) and \(R'=R \cup M\), leading to ambiguous itineraries, or thinking of \(M\) as a `sild-card' that can stand for either \(L\) or \(R\). Early on, it was realized that many of the dynamical properties of the map \(f\) are coded by the itinerary of the critical point (or the critical value), which is called the kneading sequence.

Beginning in the mid-2000's, I have used a different approach in dealing with itineraries.  If \(P=\{L,M,R]\) as above, then the quotient topology on \(P\) induced by the set of real numbers has \(L\) and \(R\) as isolated points, with \(P\) being the only neighborhood of \(M\).  Then \(i:{\rm dom}(f) \to P^\omega\) is continuous in the product topology of \(P^\omega\).  The key was realizing that if \(f\) has what I call the unique itinerary property (i.e., \(i\) is one-to-one), then the range of \(i\) is Hausdorff in the subspace topology generated by \(P^\omega\), even though \(P^\omega\) is itself obviously not Hausdorff.  Allowing \(X\) to be a dendrite instead of just an interval allowed this idea to be used to get a model for the dynamics of all quadratic Julia Sets which are also dendrites.  Later, similar arguments using infinitely many symbols allowed me to get similar results for some dendroid maps.

The spaces constructed by my original approach were all uniquely arcwise connected, which ruled out spaces containing simply-closed curves.  Later, using a similar idea, I was able to construct a similar (necessarily non-Hausdorff) topology on \(P^\omega\) (but not a product topology), using $P=\{*,#,0,1\}$,  where # and * both represented singletons which essentially acted as `wild-cards' which could stand for either 0 or 1 (with the stipulation essentially stating that * and # could not simultaneously stand for the same one), which allowed a model for Julia sets of quadratic polynomials having hyperbolic periodic points which were not \(n\)-tuplings (i.e., polynomials of the form \(f_c(z)=z^2+c\), where \(c\) was located in one of the `cardioid' of the Mandelbrot Set).  At the present, my efforts to extend these ideas to other quadratic Julia Sets have been only partly successful.