A cohomological target and periodic certificates for contact dissipation

Definitions and mathematical statement

Polterovich's Question $1.39$ conjectures that a contactomorphism $\phi$ of a closed contact manifold whose induced contact-homology action is "hyperbolic" preserves no smooth volume form, calling this strongly dissipative. The source explicitly says that hyperbolic still needs a definition. A version of contact homology, coefficients, its automorphism and the intended hyperbolicity condition must therefore be supplied before this becomes a precise implication; no meaning is silently substituted from ordinary hyperbolic dynamics.

For the aid's cooriented formulation, let $\dim M=2m+1$, choose $\alpha$ with $\xi=\ker\alpha$, and assume $\phi$ preserves coorientation, so \[ \phi^*\alpha=e^g\alpha,\qquad \phi^*(\alpha\wedge({\,\mathrm{d}}\alpha)^m) =e^{(m+1)g}\alpha\wedge({\,\mathrm{d}}\alpha)^m. \] A smooth positive volume is $e^f\alpha\wedge({\,\mathrm{d}}\alpha)^m$; it is invariant precisely when \[ (m+1)g=f-f\circ\phi. \] Equivalently $g=h-h\circ\phi$ for some smooth $h$, and this is equivalent to strictness for the conformally changed contact form $e^h\alpha$. The intrinsic obstruction is the algebraic smooth dynamical-cohomology class \[ [g]_\phi\in C^\infty(M)/\{h\circ\phi-h:h\in C^\infty(M)\}. \] Changing $\alpha$ to $e^v\alpha$ changes $g$ by $v\circ\phi-v$, so this class is contact-form independent. The undefined contact-homology conjecture is thus reduced to proving that its eventual hyperbolicity hypothesis forces $[g]_\phi\ne0$.

A sufficient test is a periodic point $x$, $\phi^N x=x$, with \[ \sum_{j=0}^{N-1}g(\phi^j x)\ne0. \] Every coboundary has zero such sum. The converse is not asserted for a general contactomorphism without additional cohomological-dynamics hypotheses.

For the normalized cotangent lift of a diffeomorphism $f:Q\to Q$, choose a Riemannian unit cosphere $S^*Q$. The lift is \[ \Psi_f(q,p)=\left(f(q),\frac{(Df_q)^{-T}p}{\|(Df_q)^{-T}p\|}\right), \qquad \Psi_f^*\alpha=\frac{\alpha}{\|(Df_q)^{-T}p\|}. \] If $q$ has period $N$ and $(Df_q^N)^{-T}$ has a real eigen-covector $p$ with eigenvalue $\lambda$, $|\lambda|\ne1$, then the corresponding unit covector is periodic for the normalized lift: after $N$ if $\lambda>0$, or after $2N$ if $\lambda<0$. Its conformal sum is respectively $-\log|\lambda|$ or $-2\log|\lambda|$, a nonzero certificate. This can happen even for a volume-preserving base diffeomorphism. The certificate is not claimed to prove that any specified contact-homology action is hyperbolic.

The source's broader theme is information retained by the smooth-to-contact constructions: a manifold $Q$ gives its unit cotangent contact manifold, a suitable embedded submanifold $K$ gives its unit conormal Legendrian, and a diffeomorphism gives a contact lift. It asks how much information is lost and refers to Eliashberg's first fifth-day question, absent here. Its final bracket asks how precisely this relates to the dissipation conjecture. That uncertainty is preserved; no general faithfulness or information-recovery theorem is supplied.

Short English statement

The source leaves contact-homology hyperbolicity undefined. The aid gives a precise target: absence of invariant volume is equivalent to a nonzero smooth conformal-factor cohomology class. Periodic conformal sums certify this, with an explicit real-eigen-covector test for cotangent lifts. The broader smooth-to-contact information question remains unspecified.

Sources

• Local UnsolvedMath registry, permanent ID $20002097$, source code catalogue entry; canonical collection aim-geometry-notes.json, notes[434].
• AIM workshop "Holomorphic curves in contact geometry," Question $1.39$, Polterovich's conjecture and theme: https://aimath.org/WWN/contactgeom2/contactgeom2.pdf. The supplied verification reports agreement of AIM HTML and PDF, with line-break repairs only; the hyperbolicity warning and bracketed questions are original. No external source or PDF was opened in this session.
• The source refers to Eliashberg's first question on the fifth day without reproducing it. No full authoritative citations for the contact-homology setup, dynamical criterion or cotangent calculation are supplied. Original remarks and literature are empty, with no proposer-field value or year.
• Supplied research attempt $1$, result "reduction," run status "open," rigor label "proved," novelty confidence "low." Registry status partially_solved, classification PARTIAL-PROGRESS, suggested difficulty L4, updated 14 August 2026.