Entropy and no escape as a convergence criterion

Definitions and mathematical statement

The extracted record is continuation prose referring to Question 24, the measures $\nu_T,\overline\nu_T$ and an unfinished rank-one sentence. None of the question, measure definitions or intended dynamical system is supplied. It records an entropy-and-no-escape strategy, not an independent complete question.

The aid sharpens that strategy into a conditional convergence statement. Let $X$ be a locally compact Polish space, $T:X\to X$ continuous, and $\mu_j$ a uniformly tight sequence of probabilities: for every $\varepsilon>0$ some compact $K$ satisfies $\mu_j(K)\geq1-\varepsilon$ for all $j$. Suppose every probability cluster point belongs to a specified class $\mathcal C$ of $T$-invariant probabilities. Assume a finite entropy ceiling $H$ on $\mathcal C$, a unique $m\in\mathcal C$ with $h_m(T)=H$, and \[ h_\mu(T)=H\quad\text{for every probability cluster point }\mu\text{ of }(\mu_j). \] Then $\mu_j\to m$ weakly as probabilities. This direct route needs no upper semicontinuity of entropy and need not define prelimit measure entropy when the $\mu_j$ are not invariant. If all prelimit measures are invariant, continuity of $T$ ensures that any probability weak limit is invariant, though membership in a narrower $\mathcal C$ remains an extra hypothesis.

The optional prelimit route instead assumes each $\mu_j$ invariant, entropy upper semicontinuity along every convergent subsequence, \[ h_\mu(T)\geq\limsup_k h_{\mu_{j_k}}(T) \quad\text{whenever }\mu_{j_k}\to\mu, \] and only the lower bound \[ \liminf_j h_{\mu_j}(T)\geq H. \] Combined with tightness, cluster-class membership and the entropy ceiling, these imply the direct cluster condition and hence convergence. Exact prelimit equality to $H$ is stronger than needed. Calling $H$ topological entropy additionally requires the appropriate definition and variational principle, especially on a noncompact space; an abstract ceiling suffices here.

For sequences, an appropriate compact exhaustion $K_R$ with $\lim_{R\to\infty}\liminf_{j\to\infty}\mu_j(K_R)=1$ supplies the no-escape condition after handling finitely many initial measures. Uniform relative compactness also gives a net version, provided one tests all convergent subnets, not only sequential cluster points. Without no escape, vague limits on $C_c(X)$ may lose probability mass, making a probability-cluster criterion vacuous. Without unique maximal entropy in the cluster class, entropy alone cannot select one limit; without the semicontinuity hypothesis, prelimit entropies cannot simply be passed to a limit. The aid mentions abstract counterexamples to these omissions, but gives no full homogeneous-dynamics application for the missing $\nu_T$ and $\overline\nu_T$.

Short English statement

The fragment proposes entropy plus no escape as a convergence strategy. The supplied aid distinguishes a direct maximal-entropy cluster criterion from a prelimit-entropy route that additionally needs semicontinuity; neither fills in the missing Question 24 data.

Sources

• Local UnsolvedMath registry, permanent ID $20003297$, source code catalogue entry; canonical aim-other-notes.json, notes[103].
• AIM workshop Emerging applications of measure rigidity: https://aimath.org/WWN/measrigid/measrigid.pdf. The record begins inside prose about Question 24 and ends "For the rank one groups,". The actual question and definitions are absent; original remarks and literature field are empty.
• Research attempt 1: context_only, problem status at run not_a_problem, current status open, classification OPEN-TRIAGE, rigor proved, novelty confidence low. No authoritative full references are supplied for the proposed abstract reduction. The machine-generated aid is not peer reviewed and requires independent verification before citation.