The three-measure Katok Finsler flow

Definitions and mathematical statement

Conjecture 9, attributed to A. Katok, says that the only ergodic probability measures for "this example" are the smooth measure and measures supported on periodic orbits. The preceding definition of the example is missing. The aid identifies the intended context as Katok's nonreversible, Liouville-ergodic 1973 Finsler geodesic-flow construction on $S^2$, with two oriented periodic orbits in phase space. Parameter choices in that construction are not specified by the extracted statement.

For a fixed such flow $\phi_t$ on its compact unit tangent or cotangent phase space, let $m$ be normalized Liouville measure and $\gamma_\pm$ its two oriented periodic phase-space orbits, of periods $P_\pm$. Their invariant probabilities are \[ \nu_\pm=\frac1{P_\pm}\int_0^{P_\pm}\delta_{\phi_t(x_\pm)}\,{\,\mathrm{d}} t. \] The target is that the ergodic invariant probabilities are exactly $m,\nu_+,\nu_-$. Equivalently, under the usual ergodic-decomposition hypotheses, every invariant probability is uniquely \[ a m+b\nu_++c\nu_-,\qquad a,b,c\geq0,\quad a+b+c=1. \] These are three measures on the flow phase space, not three atoms or necessarily three different base-surface geodesic images.

The aid's return-map reduction assumes a genuine global annular-section suspension model for the complement of $\gamma_\pm$. Let $T:A^\circ\to A^\circ$ be its interior return map and $r>0$ its return roof. In a standard finite-measure correspondence setting, for example $r$ bounded below by a positive constant, each ergodic $T$-invariant probability $\mu$ with $\int r\,{\,\mathrm{d}}\mu<\infty$ gives \[ \nu_\mu(E)=\frac{\displaystyle\int_{A^\circ}\int_0^{r(x)}\mathbf1_E(x,s)\,{\,\mathrm{d}} s\,{\,\mathrm{d}}\mu(x)}{\displaystyle\int_{A^\circ}r\,{\,\mathrm{d}}\mu}. \] The asserted three-measure classification is exactly uniqueness of the finite-mean ergodic return-map probability corresponding to $m$. Invariant return measures of infinite roof mean do not automatically give a flow probability. If a roof can approach zero, the finiteness of transverse measures needs additional care; the precise suspension hypotheses must be recovered before using this probability-normalized correspondence.

The aid also gives an independent atomicity statement: a fixed-point-free continuous flow on a compact metric space has no atomic invariant probability. An atom would have the same positive mass at every point of its nontrivial continuous orbit. In particular $\nu_\pm$ are non-atomic circle-orbit measures. A time-one-map invariant measure is a different notion and is not substituted for full flow invariance.

The literature report is qualified: Fayad--Katok prove an exact-three-measure result for carefully constructed approximation-by-conjugation transformations, and Gorodnik reports a correspondingly careful Finsler construction with precisely volume and the two closed-orbit measures. This supports the existential construction reading according to the aid, not a universal assertion for every member or an unspecified fixed member of the 1973 family. The appended Section 3.3 sentence about polygonal billiards and scarcity expectations is next-section context, not part of this conjecture.

Short English statement

The supplied aid separates a carefully constructed three-measure example from an unspecified or universal Katok-family claim. Its suspension criterion reduces the fixed-flow question to finite-roof-mean return-map uniqueness, and its periodic-orbit measures are explicitly non-atomic.

Sources

• Local UnsolvedMath registry, permanent ID $20003276$, source code catalogue entry; canonical aim-other-notes.json, notes[82].
• AIM workshop Emerging applications of measure rigidity, Conjecture 9, attributed to A. Katok: https://aimath.org/WWN/measrigid/measrigid.pdf. The supplied verification reports the intended separation from the Section 3.3 polygonal-billiard introduction. Original remarks and literature field are empty.
• The aid names Katok's 1973 Finsler construction, Fayad--Katok approximation-by-conjugation results and Gorodnik's report, but supplies no authoritative complete bibliographic entries or exact parameter choices.
• Research attempt 1: partial_result, status partially_solved, classification PARTIAL-PROGRESS, rigor proved, novelty confidence low. Its suspension reduction and atomicity lemma are machine-generated aid, not peer reviewed, and require independent verification before citation.