Jacob Palis Conjecture(Finitude of Attractors)(Dynamical Systems)

Jacob Palis Conjecture(Finitude of Attractors)(Dynamical Systems)

Definitions and mathematical statement

For a diffeomorphism $f$ of a compact manifold, let $B(A)=\{x:\operatorname{dist}(f^n(x),A)\to0\}$; for a flow replace $n$ by continuous positive time. In the selected Palis convention an attractor is a compact invariant transitive set with positive-volume basin, not necessarily an attractor defined by a trapping neighborhood. For each finite $r\ge1$, the target is density in both $\operatorname{Diff}^r(M)$ and the space of $C^r$ vector fields of systems with finitely many such sets $A_1,\ldots,A_k$ satisfying \[ \operatorname{vol}\left(M\setminus\bigcup_{i=1}^k B(A_i)\right)=0. \] The number $k$ may vary with the system. Both density assertions are required; physical measures and stochastic stability are not included in this selected part of the global conjecture.

Formulation and concept references: [S1].

Short English statement

Can every smooth dynamical system be approximated by one in which almost every initial point is attracted to one of finitely many transitive attractors?

Research attempt

Current frontier.

The incorporated definition has been checked in Palis, Sections 2.1 and 2.3, and the selected assertion is the first bullet of Section 2.7. It concerns distance convergence to compact transitive invariant sets; the subsequent requirements about physical measures and stochastic stability are not part of this target.[S1] Berger's 2025 manuscript, published in the 2026 ICM proceedings, again states the finite-attractor density question as Problem 1.4. Its discussion of generic infinitely many sinks, and its stronger parameter-family phenomena, distinguishes these from a disproof of density.[E1] Hyperbolic and one-dimensional models furnish important solved settings, but approximating every system by a uniformly hyperbolic one is not a valid general strategy.[S1][E1]

Attack route.

A hyperbolic approximation route would require more than is available in wild regions. A statistical route would have to upgrade convergence of empirical measures to actual convergence in distance; that upgrade is not automatic. The route investigated here is geometric: preserve a fixed finite collection of transitive trapping cores while enlarging its finite-time capture set. An explicit quantitative persistence lemma makes a convergent perturbation scheme rigorous provided the basin can be enlarged without adding indefinitely many cores. We then test both that proviso and a tempting genericity strengthening against examples.

Attempt.

Normalize a smooth positive volume $m$ so that $m(M)=1$. Changing this normalization has no effect on null sets.

1. Finite-time capture is a genuinely open condition. Let $f\in\operatorname{Diff}^r(M)$, and let $U_1,\ldots,U_k$ be nonempty open sets with \[ f(\overline U_i)\subset U_i,\qquad A_i(f)=\bigcap_{n\ge0}f^n(\overline U_i). \tag{205.1} \] Here $M$ is compact, so the closures are compact. Suppose, as an explicit additional hypothesis, that each $A_i(f)$ is transitive. Then $U_i\subset B(A_i(f))$: the compact sets $f^n(\overline U_i)$ are nested, and their intersection is $A_i(f)$, so their maximal distance to that intersection tends to zero. The intersection is nonempty and invariant under the homeomorphism $f$. Positive-volume basins follow since $U_i$ is nonempty and open. These are special cases of the catalogue's attractors, not a redefinition of all attractors.

Set $U=\bigcup_iU_i$ and \[ C_N(f)=\bigcup_{j=0}^{N}f^{-j}(U),\qquad C_\infty(f)=\bigcup_{N\ge0}C_N(f). \] If $m(C_\infty(f))=1$, the selected finitely many basins cover almost all of $M$. Conversely, a trapping-core argument only needs to prove this capture property, not a mixing rate or a physical measure theorem.

Lemma 205.1 (finite-capture persistence). For fixed open $U$, fixed finite $N$, and $\varepsilon>0$, the condition $m(C_N(f))>1-\varepsilon$ is open in the $C^r$ topology.

Proof. By inner regularity choose compact $K\subset C_N(f)$ with $m(K)>1-\varepsilon$. The continuous function \[ x\longmapsto \max_{0\le j\le N} \operatorname{dist}\left(f^j(x),M\setminus U\right) \] is positive on $K$ and has positive minimum $\delta$. Finite iterates of maps $g$ sufficiently close to $f$ are uniformly within $\delta/2$ of the corresponding iterates of $f$. Consequently $K\subset C_N(g)$. When $U=M$, the conclusion is immediate without the distance formula. $\square$

There is an explicit estimate behind this openness. If $f$ is $L$-Lipschitz and $\sup_xd(f(x),g(x))\le\eta$, then induction gives \[ \sup_xd(f^j(x),g^j(x)) \le\eta\sum_{a=0}^{j-1}L^a. \tag{205.2} \] Thus the above compact capture certificate is retained whenever the right side is below $\delta/2$ for $j\le N$. This estimate makes clear that long capture times may force extremely small perturbations; no uniform bound in $N$ has been smuggled into the argument.

2. What a successful diagonal perturbation would actually require. Work in a local $C^r$ Banach chart and inside a complete closed ball contained in the space of diffeomorphisms. Assume a neighborhood $\mathcal W$ has fixed $U_1,\ldots,U_k$ for which $205.1$ and transitivity hold for every system under consideration, including any limiting system in the chosen closed ball. This is a substantive persistence hypothesis.

Suppose one can choose nonempty nested closed balls $\mathcal V_n$ contained in that ball, with diameters tending to zero, and integers $N_n$ such that \[ \mathcal V_{n+1}\subset\mathcal V_n,\qquad m(C_{N_n}(g))>1-2^{-n} \quad\text{for every }g\in\mathcal V_n. \tag{205.3} \] Completeness gives a unique $g_\infty\in\bigcap_n\mathcal V_n$. For this one map, $205.3$ gives $m(C_\infty(g_\infty))\ge1-2^{-n}$ for every $n$, hence full measure. The same fixed $k$ transitive attractors from $205.1$ therefore suffice. This is a rigorous sufficient criterion; neither $k$ nor the times $N_n$ need be uniform across different initial neighborhoods.

Lemma 205.1 shows how a finite-time capture certificate at a newly chosen map can be protected in the next ball. It does not construct that map. The missing dynamical step is the following pathwise basin-completion property: inside a sequence of increasingly constrained neighborhoods, can one improve the captured measure arbitrarily close to one while retaining a finite set of trapping cores and its transitivity? The question is about the same evolving construction, not independent approximations with arbitrarily many attractors.

For vector fields take blocks satisfying $\varphi_t^X(\overline U_i)\subset\overline U_i$ for $t\ge0$ and $\varphi_t^X(\overline U_i)\subset U_i$ for $t>0$, and set \[ A_i(X)=\bigcap_{t\ge0}\varphi_t^X(\overline U_i),\qquad C_T(X)=\{x:\varphi_t^X(x)\in U\text{ for some }0\le t\le T\}. \] Forward trapping for all $t\ge0$ makes these compact sets nested; the intersection is invariant under the full flow. On a compact phase space, finite-time continuous dependence on the vector field proves the analogue of Lemma 205.1 by the same compact-certificate argument. Nested balls in $\mathcal X^r(M)$ then give the analogue of $205.3$. One must still supply the basin-completion step for vector fields separately. A perturbation of a time-one diffeomorphism need not be a time-one map of a nearby vector field, so it cannot establish this second branch.

3. An explicit smooth failure of "arbitrarily nearly full" capture. The need to control the number of cores can be tested without wild higher-dimensional dynamics. In a coordinate near $0\in S^1$, take a smooth vector field $X=v(x)\,\partial_x$ with \[ v(x)= \begin{cases} e^{-1/x^2}\sin(1/x),&x>0,\\ 0,&x=0,\\ e^{-1/x^2},&x<0. \end{cases} \tag{205.4} \] Use $205.4$ in a sufficiently small coordinate interval; choose its positive endpoint inside a positive lobe and extend $v$ positively around the remaining arc of the circle. Such an extension is smooth: one glues within positive collars, and all derivatives at $0$ vanish. It introduces no additional zeros.

For all sufficiently large $n$, the positive zeros $x_n=1/(n\pi)$ accumulating at zero satisfy \[ v'(x_n)=-(-1)^n n^2\pi^2 e^{-n^2\pi^2}. \] For even $n$ these are hyperbolic sinks; for odd $n$ they are sources. Every point in an interval between adjacent zeros flows monotonically toward its sink endpoint. Points in the complementary arc flow toward $0$ from the negative side. Thus the countably many sink singletons, together with $\{0\}$, have basins covering all points except a countable set of sources. Each is transitive and has positive-volume basin.

By continuity from below of measure, for every $\varepsilon>0$ some finite subcollection has basin union of measure exceeding $1-\varepsilon$. Nevertheless no finite collection of transitive attractors can have a full-measure basin union for this particular field. Here is a proof that does not assume attractors are minimal. If $x$ tends to a sink $s$ and also belongs to $B(A)$ for a compact invariant set $A$, then $s\in A$. If the basins of finitely many $A_i$ covered almost all of $M$, every positive-volume sink basin would force its sink to belong to some $A_i$. But a transitive set cannot contain two distinct sinks: their disjoint forward-invariant trapping neighborhoods would both have to be visited by its dense orbit, and once the orbit enters the first it cannot visit the second. This argument works even if "dense orbit" is interpreted as a two-sided orbit, since the two visit times can still be ordered. Infinitely many sinks therefore cannot be assigned to finitely many such $A_i$.

The time-one map of $205.4$ is a $C^\infty$ circle diffeomorphism with the same forward limit points and the same obstruction. Hence this failure of the approximate-coverage inference occurs in both dynamical categories in the question. It is not a counterexample to either density assertion: a single bad system is allowed.

Indeed, the vector-field example is especially easy to approximate by good systems. More generally, every $C^r$ vector field on $S^1$ can first be approximated in $C^r$ by a smooth one $v(x)\partial_x$. Choose an arbitrarily small constant $c$ such that $-c$ is a regular value of $v$. Sard's theorem in dimension one supplies such constants. Then $v+c$ either has no zeros, in which case its whole-circle periodic orbit is a transitive attractor, or has finitely many simple zeros. In the latter case the zeros alternate between sinks and sources, and all points except the sources lie in the finitely many sink basins. This proves the vector-field branch on the circle by a direct classical argument. It does not prove the higher-dimensional assertions or the general $d$-dimensional diffeomorphism branch.

4. Why replacing geometry by time averages also loses a hypothesis. There is a short exact model of the obstruction. In the two-sided shift space $\{0,1\}^{{\mathbb{Z}}}$, take $x$ with symbol $1$ at positive coordinates $2^k$ and $0$ elsewhere. For every fixed window, the set of iterates whose window contains a $1$ has asymptotic density zero. Approximation of continuous functions by finite-window functions therefore gives \[ \frac1N\sum_{j=0}^{N-1}\delta_{\sigma^j x} \ \Longrightarrow\ \delta_{0^{{\mathbb{Z}}}}. \] Yet at times $j=2^k$ the central symbol is $1$, so $d(\sigma^j x,0^{{\mathbb{Z}}})$ is bounded below in a standard product metric. Distance convergence fails. This is a logical countertest for the statistical-to-geometric inference, not a manifold counterexample or an assertion about a positive-volume family of shift points.

5. A genericity strengthening of the repair scheme is impossible. Suppose one strengthened step 2 by claiming, throughout a nonempty open set $\mathcal W$ with fixed persistent transitive trapping cores, that for every $\varepsilon>0$ the finite-time capture conditions were dense. They are open by Lemma 205.1. The Baire theorem would then make full capture by these finitely many cores a residual property in $\mathcal W$. In a Newhouse open set with residual infinitely many sinks, this contradicts the two-sink argument above. Such open sets occur in the settings described in Berger's Section 1.2.[E1] Thus an unrestricted open-dense basin-completion upgrade is not an available missing lemma. A successful density construction in those regions must be able to select an exceptional, potentially meagre set of systems rather than prove a generic finite-core theorem there.

Stress test.

The compact-certificate argument avoids unjustified interchange of limits of systems and limits of basins. It first fixes a finite time, protects the resulting compact set under perturbation, and only then takes a limit of systems. The finite family and transitivity are explicit hypotheses and have not been inferred from compactness or chain recurrence. The example $205.4$ shows why allowing that family to grow without bound is not harmless, even for smooth flows.

The argument also separates three quantifiers that cannot be exchanged: one map for all accuracy levels, a possibly different finite collection for each accuracy level, and a dense set of maps versus a residual set of maps. The Newhouse stress test rejects a stronger proposed route, not Palis's density assertion. Smooth-volume and Baire-category statements have never been identified.

\Needspace{10\baselineskip}

Outcome.

Outcome: Exploratory approach with unresolved gap.

The attempt proves finite-time capture persistence with an explicit perturbation estimate and gives a valid pathwise completion criterion for both maps and flows. A smooth circle construction disproves the inference from arbitrarily large finite basin coverage to exact finite full coverage; a separate shift example disproves the inference from statistical convergence to distance convergence. The vector-field case on $S^1$ is recovered, without a claim of novelty.

No basin-completing perturbation has been constructed for arbitrary systems while keeping finitely many transitive cores. Moreover, the most convenient generic version of that missing step is incompatible with Newhouse phenomena. Resolving the selected conjunction would need an exceptional-system construction or another mechanism that works for every finite $r$ in both categories without silently imposing trapping neighborhoods, physical measures, or hyperbolicity on the original target.

\Needspace{8\baselineskip}

Sources

• S1 J. Palis, A global perspective for non-conservative dynamics, Ann. IHP AN 22 $2005$, 485-507. https://ems.press/content/serial-article-files/15975. Catalogue formulation source.
• E1 P. Berger, Wild dynamics on manifolds, arXiv:2510.12929v1 (14 October 2025), Sections 1.1--1.2, especially Problem 1.4 and Theorem 1.7; published in the Proceedings of the International Congress of Mathematicians 2026, pp. 24--41, DOI: 10.1137/25M1810787. The genericity statements are used with their stated topology and model restrictions, not as counterexamples to density. https://arxiv.org/abs/2510.12929. https://epubs.siam.org/doi/10.1137/25M1810787.