Invariance Entropy in the Dust

Abstract: We answer negatively two natural general forms of Kawan's questions on invariance entropy for control systems, open for more than fifteen years, by a single construction. We show that finite strict invariance entropy need not coincide with ordinary invariance entropy, and that strict invariance entropy need not be lower semicontinuous under Hausdorff perturbations of the initial set. The construction is a continuous-time control system in which a Cantor coordinate stores an infinite symbolic instruction, an exponentially contracting coordinate makes late mismatches geometrically invisible, and a compact matching graph forces exact symbolic agreement. It identifies a source of information complexity not generated by dynamical expansion, but by the persistence of exact viability constraints under thin invariant geometry and by the order of limits in invariance entropy.

• The paper establishes that strict invariance entropy is positive (log2) while ordinary invariance entropy is zero, clearly delineating their separation.
• It introduces an explicit R³ nonlinear control system that employs a frozen Cantor code to demonstrate non-semicontinuity under Hausdorff perturbations.
• The study reveals that combinatorial control constraints can yield irreducible information complexity independent of traditional dynamical instability.

Invariance Entropy in the Dust: Separation and Non-semicontinuity

Introduction and Background

The paper "Invariance Entropy in the Dust" (2607.02279) addresses two long-standing open problems concerning invariance entropy in nonlinear control systems, specifically those posed by Kawan in 2009. Invariance entropy formalizes the minimal information rate required to ensure, via open-loop controls, that system trajectories remain inside a prescribed set, termed a (controlled) invariant set. Colonius and Kawan's definition—an analogue of topological entropy using control systems rather than autonomous dynamics—admits both "strict" and "ordinary" versions, which differ in whether trajectories must remain exactly or only approximately in the invariant set.

Traditionally, positive results and bounds for invariance entropy have been tied to system dynamics exhibiting instability or expansion, with rates linked to Lyapunov exponents or volume growth (e.g., [ColoniusKawan2009], [Kawan2013], [DaSilvaKawan2016Hyperbolic]). The paper constructs an explicit example demonstrating that such intuition does not capture all sources of information complexity in control tasks.

Problem Formulation

Let $(M,d)$ be a metric space for the state, $U\subset \mathbb R^m$ compact, and $\mathcal U$ the set of admissible (measurable) open-loop controls. For compact $K\subseteq Q\subset M$, the strict invariance entropy $h_{\operatorname{inv^*}}(K,Q)$ is the asymptotic exponential growth rate, as $T\to\infty$, of the minimal number of open-loop controls required to ensure trajectories starting in $K$ remain exactly in $Q$ for $[0,T]$. The ordinary invariance entropy $h_{\operatorname{inv}}(K,Q)$ relaxes this to containment in an arbitrarily small neighborhood, and $U\subset \mathbb R^m$0 holds trivially.

The critical unresolved questions are:

1. Equality for Finite Entropy: Must $U\subset \mathbb R^m$1 imply $U\subset \mathbb R^m$2?
2. (Lower Semi)Continuity under Hausdorff Approximation: Does $U\subset \mathbb R^m$3 behave lower semicontinuously with respect to Hausdorff perturbations of $U\subset \mathbb R^m$4?

Core Construction and Mechanism

The central construction is a continuous-time control system on $U\subset \mathbb R^m$5 with coordinates $U\subset \mathbb R^m$6 and control $U\subset \mathbb R^m$7:

• $U\subset \mathbb R^m$8, capturing the exponential time decay ($U\subset \mathbb R^m$9).
• $\mathcal U$0, a "frozen" coordinate storing an infinite symbolic code ($\mathcal U$1-values in the standard middle-third Cantor set $\mathcal U$2).
• $\mathcal U$3, with $\mathcal U$4 accumulating the effect of the chosen control but exponentially diminishing later contributions.

Set-up of Information Complexity:

• Initial set: $\mathcal U$5 embeds all possible Cantor codes.
• Target set: $\mathcal U$6 is the graph over $\mathcal U$7 of a function $\mathcal U$8 computed so that following the code prescribed by $\mathcal U$9 exactly results in the unique $K\subseteq Q\subset M$0 required for $K\subseteq Q\subset M$1.
• The control schedule $K\subseteq Q\subset M$2 is piecewise constant: in each unit time interval, it equals the next symbol of $K\subseteq Q\subset M$3's code.

Enforced Matching: Exact invariance (remaining in $K\subseteq Q\subset M$4) is possible if and only if the open-loop control $K\subseteq Q\subset M$5 at each unit interval matches the corresponding symbol in the code $K\subseteq Q\subset M$6 for $K\subseteq Q\subset M$7.

Fading Mechanism: Due to the exponential factor $K\subseteq Q\subset M$8, any mismatch in the control after time $K\subseteq Q\subset M$9 can at most push the trajectory away from $h_{\operatorname{inv^*}}(K,Q)$0 by an amount $h_{\operatorname{inv^*}}(K,Q)$1, i.e., its effect is geometrically negligible for approximate invariance.

Main Results and Proof Outline

1. Separation of Strict and Ordinary Invariance Entropy

The paper proves:

• $h_{\operatorname{inv^*}}(K,Q)$2
• $h_{\operatorname{inv^*}}(K,Q)$3

Mechanism: For strict invariance, every initial code in $h_{\operatorname{inv^*}}(K,Q)$4 can differ at any time, so for time horizon $h_{\operatorname{inv^*}}(K,Q)$5 at least $h_{\operatorname{inv^*}}(K,Q)$6 open-loop controls are needed, leading to positive entropy. For approximate invariance, for any $h_{\operatorname{inv^*}}(K,Q)$7, only a finite prefix of the code influences the trajectory's distance from $h_{\operatorname{inv^*}}(K,Q)$8; one can get by with $h_{\operatorname{inv^*}}(K,Q)$9 controls for $T\to\infty$0 sufficiently large so that tail errors after $T\to\infty$1 are $T\to\infty$2 in effect. Thus, the ordinary invariance entropy is zero.

2. Non-semicontinuity of Strict Invariance Entropy

A sequence of finite approximations $T\to\infty$3 to $T\to\infty$4 (using right-endpoint approximants of the Cantor set), converging in Hausdorff metric, are considered:

• For each $T\to\infty$5, $T\to\infty$6 contains only $T\to\infty$7 codes, with tail symbols fixed, thus $T\to\infty$8, as after time $T\to\infty$9 no new information enters.
• In the limit, $K$0 requires transmission of new information at every time, so $K$1.
• Therefore, $K$2; strict invariance entropy lacks lower semicontinuity under Hausdorff convergence.

3. Non-expansion Mechanism and Exact Matching

Unlike prior constructions in invariance entropy literature ([Kawan2011LowerBounds], [DaSilvaKawan2016Hyperbolic]), the positive strict invariance entropy here does not arise from local instability, volume growth, or positive Lyapunov exponents—the system's variational spectrum has rates $K$3 with no unstable direction. Instead, the essential information-theoretic obstruction is combinatorial, due to the infinite sequence of matching constraints induced by the frozen Cantor code and the geometric configuration of $K$4.

Implications and Comparison with Existing Theory

Theoretical Implications:

• Non-equivalence of invariance entropy notions: Even in the finite-entropy regime, strict and ordinary invariance entropy may diverge sharply unless additional structural conditions (such as expansion, hyperbolicity, isolation, or accessibility) hold.
• Failure of semicontinuity: Precise geometric closeness of initial sets (in the Hausdorff sense) is insufficient for lower semicontinuity of strict invariance entropy; infinite symbolic complexity can persist in the limit.
• Hidden information under approximate invariance: The ordinary invariance entropy may fail to detect persistent infinite combinatorial complexity essential for exact invariance tasks.

Practical Implications:

• Data-rate limitations: For strict invariance, 1 bit per unit time of information transmission is necessary and irreducible; for any approximate invariance with finite tolerance, a finite message suffices. This impacts quantized/numerically constrained feedback/control architectures.
• Robustness/fragility to modeling and control approximation: The result highlights the necessity of carefully distinguishing between exact and practical/approximate invariance properties in system design—practical control may have radically reduced information requirements relative to exact invariance.

Contrast with Expansion-based Examples: The appendix shows that non-semicontinuity for ordinary invariance entropy can also arise in standard expanding systems, but the mechanisms are fundamentally different: expansion amplifies initial uncertainty, whereas in the main example, strict invariance entropy is due to preserved symbolic requirements that geometric expansion cannot conceal.

Future Directions

• Identification of sharp regularity conditions: Which structural or dynamical properties guarantee equality of invariance entropies or semicontinuity with respect to set perturbations remains open.
• Development of scale-sensitive invariants: Quantities retaining dependence on both time and tolerance before taking limits may recover some of the "hidden" information overlooked in conventional invariance entropy.
• Analysis for broader classes of systems: Generalizations to stochastic systems, hybrid systems, or systems with partial observability or quantized communication constraints are natural extensions.

Conclusion

This work establishes that strict invariance entropy, even when finite, may exceed ordinary invariance entropy, and that strict invariance entropy is not, in general, lower semicontinuous in initial-set Hausdorff topology. The constructed example isolates a "zero-scale" symbolic mechanism—disconnected from dynamical instability—that produces irreducible information complexity. Consequently, classical expectations linking entropy behaviors to geometric approximation or dynamical expansion are not universally valid, emphasizing the need for refined analytical tools and conceptual clarity regarding information limitations in control theory.