Random templex encodes topological tipping points in noise-driven chaotic dynamics

Random attractors are the time-evolving pullback attractors of stochastically perturbed, deterministically chaotic dynamical systems. These attractors have a structure that changes in time, and that has been characterized recently using {\sc BraMAH} cell complexes and their homology groups. This description has been further improved for their deterministic counterparts by endowing the cell complex with a directed graph, which encodes the order in which the cells in the complex are visited by the flow in phase space. A templex is a mathematical object formed by a complex and a digraph; it provides a finer description of deterministically chaotic attractors and permits their accurate classification. In a deterministic framework, the digraph of the templex connects cells within a single complex for all time. Here, we introduce the stochastic version of a templex. In a random templex, there is one complex per snapshot of the random attractor and the digraph connects the generators or ``holes'' of successive cell complexes. Tipping points appear in a random templex as drastic changes of its holes in motion, namely their birth, splitting, merging, or death. This paper introduces and computes the random templex for the noise-driven Lorenz system's random attractor (LORA).