ω-Chaotic Maps in Topological Dynamics

• ω-chaotic maps are continuous self-maps on compact metric spaces that display uncountable ω-scrambled sets, highlighting complex orbit behavior.
• They use combinatorial properties of ω-limit sets and techniques like the Kuratowski–Mycielski lemma to construct Cantor sets with distinct dynamical characteristics.
• These maps bridge classical chaos concepts with product system dynamics, yielding examples with novel features such as zero entropy and proximality.

An $ω$-chaotic map is a continuous self-map of a compact metric space for which a specific form of sensitive dependence on initial conditions, centered on the structure of $ω$-limit sets, is present in a robust, uncountable fashion. This notion strengthens classical chaos concepts by focusing on the combinatorial richness and nontrivial intersection properties of $ω$-limit sets associated to points under iteration. The theory of $ω$-chaotic maps sits at the intersection of topological dynamics, descriptive set theory, and symbolic dynamics, providing new insight into product systems and examples with unexpected combinations of dynamical properties (Kawaguchi, 13 Jan 2026).

1. Precise Definition and Fundamental Properties

Let $(X,d)$ be a compact metric space and $f\colon X \to X$ a continuous map. Central to the theory are $ω$-limit sets: $$\omega(x,f) = \left\{ y\in X : \exists\ 0 \leq i_1 < i_2 < \cdots, f^{i_j}(x) \to y \right\}.$$ For $f$, the set of periodic points is

$\Per(f) = \bigcup_{k\geq 1} \left\{ x\in X: f^k(x)=x \right\}.$

A subset $S \subset X$ is called an $ω$-scrambled set for $f$ if for every pair of distinct $x, y \in S$:

1. $\omega(x,f)\setminus\omega(y,f)$ is uncountable.
2. $\omega(x,f)\cap\omega(y,f)\neq\emptyset$.
3. $\omega(x,f)\setminus \Per(f)\neq\emptyset$.

A map $f$ is $ω$-chaotic if it admits an uncountable $ω$-scrambled set. No further characterization of $ω$-chaos is provided beyond this precise definition, though verification often proceeds by identifying accumulation points and using cardinality arguments on $ω$-limit sets.

2. Sufficient Conditions for $ω$-Chaos in Infinite Products

A key result is a set of sufficient conditions ensuring that the infinite direct product of a map exhibits $ω$-chaos (Kawaguchi, 13 Jan 2026).

Let $f: X \to X$ be a continuous map on a compact metric space $(X,d)$. Define the infinite product space $X^{\mathbb{N}} = \prod_{n=1}^\infty X$, and set

$$g = f^{\mathbb{N}}\colon X^{\mathbb{N}}\to X^{\mathbb{N}}, \quad g\left((u_n)_{n\geq 1}\right) = (f(u_n))_{n\geq 1}.$$

Suppose there exist

• a closed set $\Lambda\subset X$, with $f(\Lambda)\subset\Lambda$,
• points $p\in \Lambda$, $z\in X$,

such that

• (a) $$\omega\bigl((p,z), f\times f\bigr)\cap\Delta \neq \emptyset$$, with diagonal $\Delta = \{(x,x): x\in X\} \subset X\times X$,
• (b) $\omega(z,f)\setminus \Lambda$ is uncountable,
• (c) $\omega(z,f)\setminus \Per(f)\neq\emptyset$,

then $g=f^{\mathbb{N}}$ is $ω$-chaotic.

These conditions package orbit convergence properties, $ω$-limit set cardinality, and the existence of nonperiodic points into a compact form, providing a mechanism for constructing new examples of chaotic behavior in product systems.

3. Techniques Underlying the Construction of $ω$-Chaotic Sets

The main proof strategy unfolds across three stages:

1. Kuratowski–Mycielski Lemma Application: Utilizing a two-point space $\{p,z\}$, construct a Cantor set $S\subset \{p,z\}^{\mathbb{N}}$ such that for any distinct $s, t \in S$, there exists $m$ with $(s_m, t_m) = (z, p)$. This ensures a form of combinatorial separation between points in $S$.
2. Shared $ω$-Limit Points: Leveraging hypothesis (a), identify a point $r$ with $(r,r) \in \omega((p,z), f\times f)$. As a consequence, $(r, r, \ldots)\in \omega(s,g)$ for all $s \in \{p,z\}^{\mathbb{N}}$, ensuring $\omega(s,g)\cap\omega(t,g)\neq \emptyset$ for any $s, t$.
3. Uncountable Set Differences: By selecting $s, t\in S$ differing at some coordinate $m$ (with $s_m = z, t_m=p$), and considering the coordinate projection $\pi_m$, the uncountability of $\omega(z,f)\setminus\Lambda$ translates into the uncountability of $\omega(s,g)\setminus\omega(t,g)$. Condition (c) further guarantees the presence of nonperiodic points.

Assembling these components, the Cantor set $S$ becomes an $ω$-scrambled set for $g$, establishing $ω$-chaos for the product map.

4. Representative Examples Highlighting New Dynamical Phenomena

The sufficient conditions admit application to construct $ω$-chaotic maps with properties not previously observed in the literature.

Example 1: Proximal, Zero-Entropy, and Not $ω^*$-Chaotic

• The base system is the unit circle $Z = S^1 \subset \mathbb{R}^2$, with the distinguished north-pole $q = (0,1)$.
• Points $Z\setminus{q} \cong \mathbb{R}$ via $i(r) = \left(2r/(r^2+1), (r^2-1)/(r^2+1)\right)$.
• The integer-shift homeomorphism $H: Z\to Z$, $H(q)=q$, $H(i(r))=i(r+1)$ is chain-transitive.
• By Bowen’s theorem, construct a compact $X$, point $x$, and homeomorphism $f$ with $f|_{\omega(x,f)} \cong H$.
• Let $Y = \{f^i(x): i\geq 0\} \cup \omega(x,f)$, $h = f|_Y$, $g = h^{\mathbb{N}}$.
• Here, $g$ is $ω$-chaotic, proximal, has zero topological entropy, and is not $ω^*$-chaotic.

Example 2: From Transitive Maps with Prescribed Mixing Properties

• For any transitive continuous map $f$ on infinite compact $X$ and $x$ with $X = \omega(x, f)$, let $Y=X$, $h=f$, $g=f^{\mathbb{N}}$.
• If $\Per(f)\neq\emptyset$ (e.g., $f$ is Devaney chaotic), $g$ is $ω$-chaotic.
• Maps constructed in [FHLO] are proximal, weakly-mixing, and uniformly rigid; their products are $ω$-chaotic, not $ω^*$-chaotic.
• Proximal, mixing maps (see Oprocha [O]) yield products that are mixing, proximal, $ω$-chaotic, but not $ω^*$-chaotic.

5. Relationship to Classical Notions and Dynamical Consequences

This approach to $ω$-chaos elucidates relationships with classical chaos frameworks:

• Flexible Construction: The main theorem enables the engineering of $ω$-chaos in infinite-product systems through a small number of orbit-limit conditions, uniting features of Li–Yorke and Devaney chaos with the more intricate $ω$-chaos paradigm.
• Novel Examples: Prior knowledge centered $ω$-chaos around systems with positive entropy or strong mixing. The existence of $ω$-chaotic, zero-entropy, and proximal systems illustrates a previously unseen breadth of dynamical behavior in such systems.
• Distinction from $ω^*$-Chaos: The constructed maps demonstrate that $ω$-chaos does not imply $ω^*$-chaos, even with significant dynamical complexity otherwise present. For example, systems can be both proximal and $ω$-chaotic, yet not $ω^*$-chaotic.

6. Open Problems and Further Research Directions

Several avenues for extension and deeper study are identified:

• Necessity of Sufficient Conditions: An open question is whether the established sufficient conditions characterize all $ω$-chaotic product systems or if sharper criteria, perhaps involving chain-recurrence or the spectral decomposition theorem, can be formulated.
• Extension to Other Dynamical Contexts: Prospective generalizations include extending the product-construction method to discrete-time semiflows, noncompact phase spaces, flows, group actions, and higher-dimensional dynamical lattices.
• Fine Structure Analysis: Investigation into the structure and dimensional properties (e.g., Hausdorff dimension) of $ω$-scrambled Cantor sets in product systems is proposed, with connections to invariant measure-theoretic properties.
• Comparison with Classical Chaos: Study of how $ω$-chaos interacts with, or diverges from, classical behaviors such as positive entropy, mixing, and rigidity in various dynamical settings.

7. Summary Table: Contrasts in $ω$-Chaotic Map Examples

Example	Key Properties	$ω$-Chaotic	$ω^*$-Chaotic	Entropy
Circle shift product	Proximal, zero-entropy	Yes	No	Zero
Product of Devaney chaotic	Mixing, positive-entropy	Yes	Yes/No*	Positive*
Proximal weakly-mixing product	Proximal, weakly-mixing, rigid	Yes	No	Variable

*Whether the product is $ω^*$-chaotic or has positive entropy depends on the detailed properties of the base map; see (Kawaguchi, 13 Jan 2026) for explicit constructions.

In summary, the theory of $ω$-chaotic maps, as established through the direct product construction, opens new possibilities in the landscape of topological dynamics, allowing for the explicit realization and characterization of $ω$-scrambled sets in settings far removed from those previously predicted by entropy or classical chaos indicators (Kawaguchi, 13 Jan 2026).