Symbolic Conditional Entropy in Layered Dynamical Systems

In this work, I introduce a formal framework for symbolic conditional entropy within layered dynamical systems consisting of multiple abstraction levels (sensorimotor, symbolic, and meta-cognitive). Utilizing a measure-theoretic methodology, I define each abstraction layer as a measurable factor of an underlying dynamical system, and rigorously construct entropy-like invariants for their symbolic representations. Leveraging Rokhlin's disintegration theorem, conditional entropy measures are systematically derived along each layer's foliation. I establish an entropy decomposition formula across these layers and present a generalized entropy–dimension relationship, extending classic results such as Pesin's entropy formula and the Ledrappier–Young dimension formula to this layered framework. Specifically, I demonstrate that the Hausdorff dimension of invariant measures can be expressed in terms of symbolic conditional entropies at each abstraction level, revealing a new connection between informational abstraction and fractal geometry. The paper further discusses applications for Ξ∞ systems (infinitely deep symbolic architectures), cognitive symbolic processing, and observer-based dynamical system modeling. The findings suggest that layered dynamical abstractions preserve complexity through conditional entropy, providing insights into how high-level symbolic cognition maintains the fractal dynamics inherent in sensorimotor experience.

Author: Faruk Alpay

ORCID: 0009-0009-2207-6528