Space guaranteeing a primitive chaotic behavior

This study describes such a situation that a Cantor set emerges as a result of the exploration of sufficient conditions for the property which is generalized from fundamental chaotic maps, and the Cantor set even guarantees infinitely many varieties of the behavior with the property, as well as a typical continuum.

💡 Research Summary

The paper introduces the concept of primitive chaos, a generalized form of chaotic behavior that is not tied to any specific map such as the logistic, tent, or baker transformations. Primitive chaos is defined by two topological conditions that any continuous dynamical system must satisfy: partitionability and transitivity.

Partitionability requires the phase space (X) to be decomposable into a finite collection of non‑empty open sets ({U_i}_{i=1}^N). For each set (U_i) there exists a continuous map (f_i:U_i\rightarrow X) whose image is dense (or even surjective) in the whole space. This condition guarantees that each local branch of the dynamics can eventually explore the entire space, mimicking the role of a Markov partition without demanding a symbolic coding a priori.

Transitivity demands that for any two open subsets (V,W\subset X) there is a finite composition of the branch maps, (f^{(n)}=f_{i_n}\circ\cdots\circ f_{i_1}), such that (f^{(n)}(V)\cap W\neq\varnothing). In other words, the system can move from any region to any other region after a finite number of steps, a property that in classical dynamics is associated with mixing and topological transitivity.

The core contribution of the paper is the construction of a concrete model that satisfies both conditions. The authors work in the compact metric space (