Dynamics of Weighted Shifts on Directed Graphs

Dimitrios Papathanasiou
Sabancı University Faculty of Engineering and Natural Sciences
 

Our recently approved TÜBİTAK 1001 project is titled “Dynamics of Weighted Shifts on Directed Graphs.” To fully understand the terms in this title — and therefore its essence — one would normally need a substantial mathematical background. In this article, however, we will attempt to illustrate the main ideas behind the project for a broader audience.

Imagine you own a hotel with ten floors, each floor containing exactly one room. If the hotel is fully booked and an additional guest arrives, there is no way to accommodate this guest without asking someone else to leave. Now imagine instead that your hotel has infinitely many floors. Let us repeat the same scenario: the hotel is fully booked, and an extra guest requests a room. Surprisingly, this time the problem has a simple solution. You can ask each current guest to move one floor up — the guest on the first floor moves to the second, the guest on the second moves to the third, and so on. In this way, every existing guest still has a room, and the first floor becomes available for the new arrival.

This thought experiment is famously known as Hilbert’s Hotel, and it serves as an illustration of what mathematicians call a shift operator. Now let us enrich the picture. Suppose we assign a number to each guest in Hilbert’s Hotel. When a guest moves one floor up, we multiply their assigned number by the number of their new floor. This gives us an example of a weighted shift. For instance, if the guest on the 9th floor was assigned the number 3, then after moving to the 10th floor, the assigned number becomes 10×3=30.

Weighted shifts play a central role in Functional Analysis, a branch of mathematics concerned — loosely speaking — with infinite-dimensional vector spaces and the linear operators acting on them. Over time, mathematicians realized that it is fruitful to enlarge the class of weighted shifts while preserving some of their desirable properties. This led to the study of weighted shifts not only on simple structures like the infinite hotel, but on trees and, more generally, on directed graphs. The underlying idea remains similar: instead of shifting numbers along the floors of Hilbert’s Hotel, imagine shifting them along the vertices of a binary tree. As the graph becomes more intricate — with multiple branching points or even loops — the corresponding shift operator becomes increasingly complex.

Having gained some intuition about weighted shifts, the next question is: which of their properties are we interested in studying? To answer this, we turn to the theory of dynamical systems.

A dynamical system consists of a set — whose elements represent possible states — together with an evolution function. As the name suggests, the evolution function describes how states change over time. Starting from a given state, applying the evolution function once tells us the state after one unit of time. Applying it twice tells us the state after two time units, and so on. These repeated applications are called iterations, and the sequence of states they generate is known as the orbit of the initial state. Orbits are central to understanding the long-term, or asymptotic, behavior of the system.

A dynamical system is called chaotic if, roughly speaking, its long-term behavior is unpredictable. One of the key features of chaos is sensitive dependence on initial conditions: two states that start out very close to each other may eventually evolve into states that are vastly different. This phenomenon was famously illustrated by Edward Lorenz through the so-called butterfly effect: the flap of a butterfly’s wings in one part of the world may, through a chain of amplifications, ultimately contribute to the formation of a tornado elsewhere.

In our project, we study weighted shifts from the perspective of dynamical systems, viewing them as evolution maps. Our main objective is to understand when such shifts exhibit chaotic behavior. At first glance, this may seem paradoxical. Weighted shifts are linear operators, and linear systems are often associated with predictability and simplicity. Yet, many of these operators do exhibit chaotic dynamics. The key reason lies in the fact that they act on infinite-dimensional spaces, where intuition drawn from finite-dimensional settings no longer applies.

We are excited to begin work on this three-year project. The research team — soon to be formed — will consist of two Master’s students, one PhD student, and a postdoctoral researcher, in addition to myself. We look forward to exploring these questions together and contributing to the growing understanding of dynamics of weighted shifts on directed graphs.