## 5835 - The maps of time

### N. Lygeros

Translation: Paola Vagioni

Via geography, we have combined in an exclusive way the notion of the map with that of space. And while we are having difficulties in comprehending time, we do not realize that our first admittance allows for the existence of a map of time. In reality, in mathematics, while time is an imaginary dimension of space-time, we can record it on a map. Indeed, we also have the section of Poincaré which records the orbital systems whenever there is a difficulty in formulating the orbit. More generally, in fractal analysis we make an intensive use of the parameters plane which represents a simple form of the notion of a map of time. For example, the Mandelbrot set is considered as a map of the Julia sets when these have the capacity of cohesion. In consequence, we have at our disposal a mathematical tool, which can be interpreted as the substratum of the notion that we examine. In order to apply it in the area of strategy, we ought to relieve ourselves from geostrategy, which is by nature related to the idea of geometry. Via topostrategy, which is based on the relationships between entities and not the entities themselves, we can construct in an effective way the time map. For this reason, we must record the relationships inside time and space. Moreover, by doing a structure transference, we can create the equivalent of the Voronoi diagrams. Only this time distance will not be spatial but temporal. In this way we can examine the robustness of the structures and better comprehend the notions of alliance and enemy. Because with the spatial system, we have a simplistic expression of game theory since the horizon is the present, which means that it exists without repercussions for the game in the future. In other words, it is always about one game. While with the absence of a predetermined horizon, the game becomes more complex and demands the use of strategy that lasts over time. For the possibilities of this approach to become more understandable, it suffices to take the example of the Byzantine Empire. In the space of time, regarding the terrestrial conquests, the Byzantine has changed so much that we are obliged to determine its time period. Moreover, we do not pay attention in this way to its great duration. While in the topostrategic and time map approaches, we can acquire a picture which better depicts the real structure of the Byzantine Empire. In consequence, we can also comprehend the strategy that it used all these centuries since the maps of time will also demonstrate its depth.