The categorical basis of dynamical entropy

Abstract

Many branches of theoretical and applied mathematics require a quantifiable notion of complexity. One such circumstance is a topological dynamical system - which involves a continuous self-map on a metric space. There are many notions of complexity one can assign to the repeated iterations of the map. One of the foundational discoveries of dynamical systems theory is that these have a common limit, known as the topological entropy of the system. We present a category-theoretic view of topological dynamical entropy, which reveals that the common limit is a consequence of the structural assumptions on these notions. One of the key tools developed is that of a qualifying pair of functors, which ensure a limit preserving property in a manner similar to the sandwiching theorem from Real Analysis. It is shown that the diameter and Lebesgue number of open covers of a compact space, form a qualifying pair of functors. The various notions of complexity are expressed as functors, and natural transformations between these functors lead to their joint convergence to the common limit.

Figures (2)

• Figure 1: Outline of the categorical approach. Section \ref{['sec:intro']} describes the three classical ways of quantifying the growth of complexity of a dynamical system. We interpret each of these notions of complexity as functors $\mathop{\mathrm{ Dyn }}\nolimits$\ref{['eqn:def:dyn']}, $\mathop{\mathrm{ sep }}\nolimits$\ref{['eqn:def:OrbCovSep']} and $\mathop{\mathrm{ span }}\nolimits$\ref{['eqn:def:OrbCovSep']}. We have labelled these as the dynamics induced functors. We define a category of growth rates denoted $\mathcal{Z}$\ref{['eqn:def:rates']} and show that the three dynamics induced functors lead to entropy functors $\mathop{\mathrm{\mathcal{P}}}\nolimits_1$\ref{['eqn:def:P1_P4_fnctr']} and $\mathop{\mathrm{\mathcal{P}}}\nolimits_2, \mathop{\mathrm{\mathcal{P}}}\nolimits_3$\ref{['eqn:def:_P2_P3_fnctr']}. These are the functorial representation of the sequences described in Equation \ref{['eqn:pressure']}, which describe how the classical notions of complexity grow with successive iterations of the dynamical system. The asymptotic exponential growth rate of these sequences are interpretable as the colimit functor, and we derive a general categorical result (Theorem \ref{['thm:entangle']}) to show that they are equal.
• Figure 2: Entanglement network between entropy functors. The figure shows a graph on 7 vertices, with each vertex drawn as a box. For each $1\leq i \leq 7$, the $i$-th vertex represents a functor $F_i : C_i \to \mathcal{Z}$, from some category $C_i$ into the same codomain $\mathcal{Z}$, which is the rate category \ref{['eqn:def:rates']}. Each such functor represents a functorial / structural way of assigning a complexity sequence to a topological dynamical system $f:\Omega\to \Omega$. Each edge from a vertex $i$ to a vertex $j$ represents a sub-structural relationship between $F_i$ and $F_j$, made precise in Theorem \ref{['thm:entangle']}. The existence of such an edge implies that the colimit of $F_i$ is less than or equal to the colimit of $F_j$. Since this entangle network is strongly connected, all these colimits must be equal. These colimits are related to the classical entropy limits expressed in \ref{['eqn:pressure']}, and thus they are proved to be equal.

Theorems & Definitions (16)

• Corollary 1: Topological entropy
• Lemma 2.1
• Lemma 2.2
• Theorem 2: Qualifying pair construction
• Theorem 3: Qualifying action
• Corollary 4: Diameter-Lebesgue num. pair
• Lemma 3.1
• Lemma 4.1
• Lemma 4.2
• Theorem 5: Limits under qualifying pair