Generic coincidence of Milnor, statistical, and physical attractors beyond the continuous interval setting

For a self-map $f$ of a compact space $X$ with reference measure $\lambda$ (Lebesgue measure when $X$ is a manifold), the global Milnor attractor $\Lambda(f)$, the statistical attractor $A_{stat}(f)$, and the physical (minimal) attractor $A_{phys}(f)$ formalize increasingly weak aspects of the asymptotic behavior observable from $\lambda$-typical initial conditions, and always satisfy $A_{phys}(f)\subseteq A_{stat}(f)\subseteq\Lambda(f)\subseteq\Omega(f)$, where $\Omega(f)$ is the non-wandering set. Whether all such attractor notions coincide for Baire-generic dynamical systems — Ilyashenko's Problem 1 of 2008, raised earlier in the monograph of Sharkovsky, Kolyada, Sivak and Fedorenko — was answered affirmatively for the space $C([0,1])$ of continuous interval maps with the uniform topology by Foryś-Krawiec, Hantáková, Kowalewski and Oprocha (2025), who moreover showed the common attractor equals $\Omega(f)$, is a zero-Hausdorff-dimensional Cantor set, and is robust but not Lyapunov stable. The surviving general form of the problem is: determine for which function spaces and topologies a residual set of maps has coinciding Milnor, statistical, and physical attractors — for instance for $C^r$ interval maps with $r\ge 1$, for maps and homeomorphisms of compact manifolds of dimension at least two, and for continuous maps of other one-dimensional continua such as the circle and finite graphs. No resolution of any of these settings is known in the literature to date, so the problem remains open.

Background

For a discrete-time dynamical system $f:X\to X$ on a compact phase space, the asymptotic behavior of a single orbit is recorded by its $\omega$-limit set $\omega(x)$, the set of accumulation points of the orbit of $x$. Taking $\lambda$ to be a reference measure on $X$ (Lebesgue measure when $X$ is a manifold), one can ask which part of the dynamics is visible from $\lambda$-almost every initial condition. Milnor's global attractor $\Lambda(f)$ is the smallest closed set containing $\omega(x)$ for $\lambda$-almost every $x$. Because an orbit may spend asymptotically negligible time near parts of its $\omega$-limit set, the bifurcation-theory literature (Arnol'd, Afraimovich, Ilyashenko and Shilnikov, in the Encyclopaedia volume Dynamical Systems V) introduced the statistical attractor $A_{stat}(f)$: the smallest closed set outside every neighborhood of which a $\lambda$-typical orbit spends asymptotically almost no time, equivalently the closed set generated by the statistical ($\sigma$-)limit sets that record points whose neighborhoods are visited with positive asymptotic frequency. Ilyashenko's physical, or minimal, attractor $A_{phys}(f)$ lies below even this: it is the smallest closed set containing the supports of all weak-$*$ limit points of the empirical measures $\frac{1}{n}\sum_{i<n}f^i_*\lambda$, that is, of all invariant statistics generated by evolving the reference measure. (This is a set-valued notion, not to be confused with the support of a physical, or SRB, measure — an invariant measure whose basin has positive $\lambda$-measure — and such a measure need not exist.) For every continuous $f$ one has the chain of inclusions $A_{phys}(f)\subseteq A_{stat}(f)\subseteq\Lambda(f)\subseteq\Omega(f)$, where the non-wandering set $\Omega(f)$ consists of the points every neighborhood of which returns under some iterate; it records all recurrent topological dynamics, including dynamics supported on sets of $\lambda$-measure zero. The three observable attractors need not coincide for an individual system: Kleptsyn constructed a system whose minimal (physical) attractor is strictly smaller than its statistical attractor, and Foryś-Krawiec, Hantáková, Kowalewski and Oprocha give examples with strict inclusions $A_{phys}\subsetneq A_{stat}\subsetneq\Lambda$, as well as a map whose Milnor attractor has zero topological entropy while a disjoint zero-measure Cantor set carries all the entropy of the system.

The question whether the notions nonetheless agree for typical systems is classical. It was raised in the monograph of Sharkovsky, Kolyada, Sivak and Fedorenko on one-dimensional dynamics and stated as Problem 1 in Ilyashenko's 2008 list of open problems in Nonlinearity: "Is it true that, for generic dynamical systems, all the various types of attractors coincide?" Here generic is meant in the sense of Baire category: a property holds generically in a function space — such as $C([0,1])$ with the topology of uniform convergence, or a $C^r$ space with its standard topology — if it holds on a residual set, a countable intersection of open dense sets. Before 2025 only partial answers were known. Okunev proved that for a generic orientation-preserving partially hyperbolic skew product with circle fibers over an Anosov diffeomorphism (a uniformly hyperbolic base map), the Milnor and statistical attractors coincide and are Lyapunov stable (a set is Lyapunov stable if orbits starting arbitrarily close to it remain close to it forever); the physical attractor was not treated. Abdenur and Andersson showed that a generic homeomorphism or continuous map of a compact manifold of dimension at least two has no physical measure at all, even though the Birkhoff time averages $\frac{1}{n}\sum_{i<n}\varphi(f^i(x))$ of continuous observables converge for $\lambda$-almost every $x$ — so no single invariant measure describes the typical statistics; this sharpens the landscape without settling the comparison of the set-valued attractors. Pinheiro proved that for every fixed non-degenerated $C^2$ interval map (finitely many non-flat critical points) the topological and statistical attractors coincide for a Baire-generic set of points of the interval — a genericity statement in phase space rather than in the space of maps.

Foryś-Krawiec, Hantáková, Kowalewski and Oprocha resolved the original one-dimensional continuous case. Foryś-Krawiec, Hantáková, Kowalewski and Oprocha prove that there is a residual set $R\subset C([0,1])$ such that every $f\in R$ satisfies $\Lambda(f)=A_{stat}(f)=A_{phys}(f)=\Omega(f)$: for a Baire-typical continuous interval map, the asymptotic dynamics detected by Lebesgue-almost every initial condition recovers all recurrent dynamics. The common attractor is a Cantor set of zero Hausdorff dimension; it is robust (the map $g\mapsto\Lambda(g)$ is continuous at $f$ for the Hausdorff metric) but not Lyapunov stable, and every invariant probability measure of such an $f$ has a basin of zero Lebesgue measure, so no physical measure and no Palis attractor (a transitive attractor with a basin of positive Lebesgue measure, in the sense used in Palis's finiteness conjecture for typical smooth parameter families) exists. The proof is intrinsically one-dimensional and purely topological: it builds on Mizera's description of generic continuous interval maps — for a residual set of maps, $\lambda$-almost every orbit has a solenoid as its $\omega$-limit set (a nested family of closed intervals mapped cyclically with unboundedly many components) and is governed by an adding machine, that is, a translation on an inverse limit of finite cyclic groups — and it simultaneously controls the periodic points created during the approximation. None of this machinery survives unchanged in higher regularity, in higher dimensions, or even on the circle, and the paper explicitly presents its result as the answer in the $C([0,1])$ setting, leaving the general form of Problem 1.1 open. (The same paper also contrasts its conclusions with Palis's conjecture on the finiteness of attractors for generic smooth parameter families and notes that robustness of $\Lambda$ without Lyapunov stability, which it exhibits in the $C^0$ interval category, is unexplored in other categories.)

Problem Statement

Let $X$ be a compact metric space with a reference Borel measure $\lambda$ (Lebesgue or volume measure when $X$ is a manifold), and let $f:X\to X$ be a continuous — or, in the smooth instances, $C^r$ — self-map. Define: the global Milnor attractor $\Lambda(f)$, the smallest closed set containing $\omega(x)$ for $\lambda$-almost every $x$; the statistical attractor $A_{stat}(f)$, the smallest closed set outside every neighborhood of which a $\lambda$-typical orbit spends asymptotically almost no time; and the physical (minimal) attractor $A_{phys}(f)$, the smallest closed set containing the supports of all weak-$*$ limit points of the empirical measures $\frac{1}{n}\sum_{i<n}f^i_*\lambda$. These always satisfy $A_{phys}(f)\subseteq A_{stat}(f)\subseteq\Lambda(f)\subseteq\Omega(f)$, where $\Omega(f)$ is the non-wandering set. The case $X=[0,1]$ with the space $C([0,1])$ of continuous maps in the uniform topology has been resolved positively (a residual set of maps has $A_{phys}=A_{stat}=\Lambda=\Omega$).

The problem, in the general form of Ilyashenko's Problem 1 ('Is it true that, for generic dynamical systems, all the various types of attractors coincide?'), is: determine for which function spaces and topologies a residual set of maps satisfies $A_{phys}(f)=A_{stat}(f)=\Lambda(f)$. Natural instances of this general question in which no resolution is currently known include $C^r([0,1])$ with the $C^r$ topology for $r\ge 1$; spaces of continuous maps and of homeomorphisms of compact manifolds of dimension at least $2$, in the $C^0$ and $C^r$ topologies; and spaces of continuous maps of other one-dimensional continua, such as the circle and finite graphs, in the uniform topology. These settings are illustrative instances of the unqualified problem, not a mandatory checklist that an answer must exhaust. For a given function space and topology, a resolution may be positive — a residual set on which all three attractors coincide — or negative — a residual set on which at least one of the inclusions $A_{phys}\subseteq A_{stat}\subseteq\Lambda$ is strict. The stronger identities proved in the $C([0,1])$ case, namely equality of the common attractor with $\Omega(f)$ and robustness of $\Lambda$ in the Hausdorff metric, are natural companion questions but are not required for resolution.

References

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