Events for 01/22/2025 from all calendars
Groups and Dynamics Seminar
Time: 3:00PM - 4:00PM
Location: BLOC 628
Speaker: Zoran Sunik, Hofstra University
Title: Title: On the conjugator between the Collatz map and the shift map
Abstract: Certain parity functions
were used by Terras in his study of the Collatz map. One can combine these into a single function, call it gamma, understood as an isometry (and hence measure preserving bijection) of the ring Z_2 of dyadic integers. The map gamma conjugates the Collatz map T (extended to Z_2) to the one-sided shift map on Z_2 and, thus, T is ergodic on Z_2. Lagarias related properties of gamma to the Collatz conjecture and other conjectures associated to it. For instance, the Collatz Conjecture is equivalent to the claim that gamma(Z^+) is a subset of 1/3*Z, while the conjecture that all orbits of the Collatz map on Z are eventually periodic is equivalent to the claim that gamma(Z_2 intersected with Q) = Z_2 intersected with Q. Bernstein and Lagarias provided some results on the cycle structure
of gamma and, based on these results and empirical observations, they stated several conjectures.
We revisit the study of gamma, but think of it as a binary tree automorphism. In fact, we start with a more general approach. Namely, if alpha_0 and alpha_1 are two binary tree automorphisms and T is defined by cases, as \alpha_0(u/2) for even u and alpha_1((u-1)/2) for odd u in Z_2, that is, for all w, T(0w)=alpha_0(w) and T(1w)=alpha_1(w), we define gamma as the unique tree automorphism that conjugates T to the shift map and preserves parity. We prove that, if both alpha_0 and alpha_1 are finite state automorphisms (which they are in case of T), then gamma(Z_2 intersected with Q) contains Z_2 intersected with Q. Moreover, if the minimal self-similar group generated by alpha_0 and alpha_1 is contracting then we have equality, that is, gamma(Z_2 intersected with Q) = Z_2 intersected with Q.
In the concrete case when gamma is the conjugator of the Collatz map, by using the tools and the language of tree automorphisms we reprove the results of Bernstein and Lagarias and provide some further insights into gamma. We end with some new empirical data and open questions. In particular, we provi