Chaos of Individual Operators in Chaotic Semigroups

Definitions and mathematical statement

Let $E$ be a separable complex Banach space and $(T_t)_{t\geq0}$ a $C_0$-semigroup of bounded linear operators: $T_0=I$, $T_{s+t}=T_sT_t$, and $t\mapsto T_tx$ is norm-continuous for every $x\in E$. Under the standard Devaney convention, the semigroup is chaotic if it has a dense orbit $\{T_tx:t\geq0\}$ and its periodic-point set \[ \operatorname{Per}(T)=\bigcup_{s>0}\ker(T_s-I) \] is dense in $E$.

For a fixed $t>0$, the individual bounded operator $T_t$ is chaotic if some discrete orbit $\{T_t^nx:n\geq0\}=\{T_{nt}x:n\geq0\}$ is dense and \[ \operatorname{Per}(T_t)=\bigcup_{m\geq1}\ker(T_{mt}-I) \] is dense. Thus the semigroup permits all positive periods, whereas the individual operator uses periods that are integer multiples of one fixed time. Dense semigroup periodic points do not by definition establish density of each fixed-time subset.

The record poses two distinct assertions for every chaotic semigroup in this class: \[ \forall t>0:\ T_t\text{ is chaotic}; \qquad\text{and, more weakly,}\qquad \exists t_0>0:\ T_{t_0}\text{ is chaotic}. \] The time in the existential version may depend on the semigroup. Neither question is reduced to mere hypercyclicity, and the $t=0$ identity is excluded as stated.

The supplied literature triage checked 21 August 2026 reports no authoritative resolution of the extracted questions in its targeted search. It supplies no separate theorem or counterexample and is retained as dated source metadata rather than an independent current-status conclusion.

Short English statement

If a strongly continuous semigroup has a dense orbit and dense periodic points, must every positive-time operator be chaotic, or at least one? Continuous-time periods and integer multiples of a fixed time are different requirements. Both strength levels are preserved.

Sources

• Local UnsolvedMath registry, permanent ID 30000576, exact source code catalogue entry; Chaos of Individual Operators in Chaotic Semigroups. Proposed year 2006, status open; corrected_verified extraction preserves both universal and existential questions on separable complex Banach spaces.
• Supplied report citation: Mini-Workshop: Hypercyclicity and Linear Chaos (2007), Oberwolfach Reports, DOI 10.4171/owr/2006/37, pp. 2227--2276, https://doi.org/10.4171/owr/2006/37. Title year differs from DOI/proposed year.
• Supplied literature triage checked 2026-08-21 cites the same report, https://doi.org/10.4171/owr/2006/37, and reports no authoritative resolution found. The dated OWR-derived curation requests independent verification of time-sensitive status and cited claims.