Barrientos and Rojas (Math. Z. 2023) proved that the coexistence of infinitely many non-hyperbolic (Hénon-like) strange attractors is Kolmogorov typical in sectional-dissipative Berger domains of kk-parameter families of diffeomorphisms of dimension m≥3m\geq 3 with 3≤d<r−13\leq d<r-1, answering a 1998 question of Colli. Their result leaves two linked questions open. First, the two-dimensional case: Palis conjectured that for generic families of one-dimensional dynamics and of surface diffeomorphisms, the parameter set carrying infinitely many coexisting attractors has Lebesgue measure zero, while all known counterexamples (the Berger phenomenon, including the germ-typical Newhouse phenomenon of Berger–Crovisier–Pujals) rely on mechanisms — blenders, parablenders, heterodimensional cycles — that are structurally unavailable for invertible surface diffeomorphisms. Second, the regularity boundary: it is unknown whether typical coexistence of infinitely many strange attractors extends past the hypothesis d<r−1d<r-1, in particular to C∞C^\infty parameter dependence, a frontier that Berger's 2025 survey confirms is open even for infinitely many sinks. Both components remain open as of 2025.

Background

A discrete dynamical system given by a diffeomorphism ff of a compact manifold MM can have several attractors simultaneously. A (non-hyperbolic) strange attractor is a compact invariant set Λ\Lambda whose basin of attraction has non-empty interior and which contains a point with dense orbit and exponentially growing derivative along its iterates; the Hénon attractor is the classical example. A hyperbolic periodic point QQ has a homoclinic tangency when the unstable and stable invariant manifolds of the orbit of QQ meet non-transversally; unfolding such a tangency in a family of maps is a primary mechanism for creating chaotic dynamics. A periodic point is sectional dissipative when the product of the moduli of any two of its multipliers is less than one (so areas in unstable directions contract). Newhouse showed that the set of C2C^2 surface diffeomorphisms with homoclinic tangencies has non-empty interior (a Newhouse domain), and that a generic diffeomorphism in such a domain has infinitely many coexisting sinks (attracting periodic orbits) — the Newhouse phenomenon. Mora–Viana and Viana proved that Hénon-like families — perturbations of the quadratic family occurring when a sectional-dissipative tangency is unfolded — exhibit a strange attractor for a positive-Lebesgue-measure set of parameters. Colli (1998) and Leal (2008) obtained dense (but not measure-persistent) coexistence of infinitely many strange attractors in Newhouse domains, and Colli asked whether 'most' families display this coexistence persistently.

To speak of 'most' families one uses Kolmogorov typicality: a property PP is Cd,rC^{d,r}-Kolmogorov typical in an open set UU of kk-parameter CdC^d-families f=(fa)a∈Ikf=(f_a)_{a\in I^k} (where IkI^k is a cube of parameters) of CrC^r-diffeomorphisms if a Baire-residual set of families in UU has PP for faf_a at Lebesgue-almost-every parameter aa. Barrientos–Rojas introduced (with Raibekas) the notion of a Cd,rC^{d,r} Berger domain: an open set UU of families in which a dense subset of families has, around each parameter ball of a covering, a homoclinic tangency persisting (and moving continuously) with the parameter — the parametric analogue of a Newhouse domain. In such domains Berger had already proved the typical coexistence of infinitely many sinks (the Berger phenomenon) for surface endomorphisms (local diffeomorphisms) and for diffeomorphisms of dimension at least three, using parablenders — parametric analogues of blenders, which are hyperbolic sets whose unstable lamination robustly covers an open set; Berger–Crovisier–Pujals (2022) strengthened this to germ-typicality near a 'bicycle' (a dissipative homoclinic tangency linked to a heterodimensional cycle, i.e. a cycle between periodic points of different unstable dimensions). Barrientos–Rojas (2023) then proved Theorem A: in sectional-dissipative Cd,rC^{d,r} Berger domains of families of diffeomorphisms of dimension m≥3m\geq 3 with 3≤d<r−13\leq d<r-1, the coexistence of infinitely many non-hyperbolic strange attractors is Kolmogorov typical — the non-robustness of strange attractors is circumvented by combining Berger's paratangency scheme with the prevalence (uniform positive-measure) of strange attractors in Hénon-like renormalizations, and the constraint d<r−1d<r-1 comes from the parametrized rescaling lemma that controls derivatives with respect to the parameter.

This answers Colli's question only in dimension m≥3m\geq 3 and finite parameter regularity. In the opposite direction, Palis conjectured (see Palis 2000) that for a generic family of one-dimensional dynamics or of surface diffeomorphisms, the set E(f)E(f) of parameter values where infinitely many attractors coexist has Lebesgue measure zero. For interval maps with finitely many non-flat critical points a single map has only finitely many attracting cycles (each attracts a critical point), so the substantive open case is that of invertible surface diffeomorphisms. There the known mechanisms for the Berger phenomenon fail structurally: surface diffeomorphisms admit no heterodimensional cycles (all periodic points have the same index) and no blenders of the required kind, and the best partial result — Berger–de Simoi (2016) — shows only that during the one-dimensional unfolding of a dissipative quadratic homoclinic tangency of a surface diffeomorphism the infinitely-many-sinks parameter set has Hausdorff dimension at least 1/21/2, far from positive Lebesgue measure or genericity in the space of families. Berger's 2025 survey states that Palis' vision 'remains strong in the category of surface diffeomorphisms' and poses as an open problem even the existence of a single open set of C∞C^\infty families whose infinitely-many-attractor parameter set has positive measure, the finite-regularity counterpart being exactly what the known theorems settle.

Problem Statement

Determine the low-dimensional and regularity scope of the Berger phenomenon, in both directions, with the following two linked components.

(A) The surface case of Palis' parameter-measure conjecture. For kk-parameter Cd,rC^{d,r} families f=(fa)a∈Ikf=(f_a)_{a\in I^k} (where IkI^k is a cube of parameters) of CrC^r diffeomorphisms (r≥2r\geq 2) of a compact surface, decide which of the following holds: (i) Palis' conjecture is true — for a Baire-generic set of families, the set E(f)={a:fa exhibits infinitely many coexisting attractors}E(f)=\{a : f_a \text{ exhibits infinitely many coexisting attractors}\} has Lebesgue measure zero (the conjecture as originally stated covers also families of one-dimensional dynamics; for interval maps with finitely many non-flat critical points a single map has only finitely many attracting cycles, so the substantive case is the surface one); or (ii) Palis' conjecture fails — there exists an open set UU of Cd,rC^{d,r} families of surface diffeomorphisms such that, for a non-meager (in particular, Baire-residual) subset of families in UU, the set E(f)E(f) has positive Lebesgue measure, the infinitely many coexisting attractors being hyperbolic sinks or non-hyperbolic Hénon-like strange attractors. The known Berger-phenomenon counterexamples (for surface endomorphisms and for diffeomorphisms of dimension m≥3m\geq 3) take the stronger form of a sectional-dissipative Cd,rC^{d,r} Berger domain — an open set of families with dense, parameter-persistent homoclinic tangencies in the sense of Barrientos–Rojas Definition 1.1 — in which the coexistence of infinitely many attractors is Kolmogorov typical, i.e. occurs for Lebesgue-almost-every parameter of every family in a residual subset of the domain; exhibiting such a domain of surface-diffeomorphism families would establish (ii), but is not required for it: an open set of surface families whose infinitely-many-attractor parameter sets have positive Lebesgue measure already refutes the conjecture (compare Problem 1.10 of Berger's 2025 survey, posed for C∞C^\infty families).

(B) The regularity boundary of typical coexistence. Decide whether the Barrientos–Rojas theorem extends beyond the hypothesis d<r−1d<r-1: in sectional-dissipative Cd,rC^{d,r} Berger domains of kk-parameter families of diffeomorphisms of dimension m≥3m\geq 3, is the Kolmogorov-typical coexistence of infinitely many non-hyperbolic strange attractors still valid when d≥r−1d\geq r-1 — in particular for d=r−1d=r-1, for d=rd=r, or for C∞C^\infty parameter dependence (the case posed as open by Berger's 2025 survey even for infinitely many sinks)? The quantified class is nonempty at the finite boundary regularities: Berger constructed sectional-dissipative Cd,rC^{d,r} Berger domains of persistent homoclinic tangencies with d≤rd\leq r for diffeomorphisms in higher dimension as well as for surface endomorphisms, and the Cd,rC^{d,r} spaces of families and the notion of Kolmogorov typicality are defined for 0<d≤r≤∞0<d\leq r\leq\infty (Barrientos–Rojas §2.1); for C∞C^\infty parameter dependence, the existence of such domains is itself part of the open frontier (Berger's Conjecture 1.8 expects it even for sinks). A negative answer at a boundary regularity d≥r−1d\geq r-1 would exhibit a sectionally dissipative Berger domain at that regularity on which the Kolmogorov-typical coexistence guaranteed by the Barrientos–Rojas theorem at regularities d<r−1d<r-1 genuinely fails.

The two components are two faces of one scientific objective: to delimit, by dimension and by parameter regularity, the region of the space of parametric families in which persistent coexistence of infinitely many attractors is Kolmogorov typical, and thereby to settle the last standing low-dimensional case of Palis' conjecture on the measure of infinitely-many-attractor parameter sets.

References

  1. Pablo G. Barrientos and Juan David Rojas, Typical coexistence of infinitely many strange attractors, Mathematische Zeitschrift 303(2), article 34 (2023). DOI: 10.1007/s00209-022-03183-5; arXiv:2102.08316. https://doi.org/10.1007/s00209-022-03183-5; https://arxiv.org/abs/2102.08316
  2. Pierre Berger, Sylvain Crovisier and Enrique Pujals, Germ-typicality of the coexistence of infinitely many sinks, Advances in Mathematics 407, article 108528 (2022). DOI: 10.1016/j.aim.2022.108528; arXiv:2103.16697. https://doi.org/10.1016/j.aim.2022.108528
  3. Pierre Berger, Generic family with robustly infinitely many sinks, Inventiones mathematicae 205(1), 121–172 (2016). DOI: 10.1007/s00222-015-0632-6; arXiv:1411.6441. https://doi.org/10.1007/s00222-015-0632-6
  4. Pierre Berger, Wild dynamics on manifolds, Proceedings of the International Congress of Mathematicians 2026, Volume 5: Invited Lectures (Sections 9-11), SIAM, 24-41 (2026); preprint arXiv:2510.12929 (2025). DOI: 10.1137/25m1810787. https://arxiv.org/abs/2510.12929; https://doi.org/10.1137/25m1810787
  5. Pierre Berger and Jacopo De Simoi, On the Hausdorff dimension of Newhouse phenomena, Annales Henri Poincaré 17(1), 227–249 (2016). DOI: 10.1007/s00023-015-0404-2. https://doi.org/10.1007/s00023-015-0404-2
  6. Pablo G. Barrientos and Artem Raibekas, Berger domains and Kolmogorov typicality of infinitely many invariant circles (2021). arXiv:2102.07962. https://arxiv.org/abs/2102.07962
  7. Eduardo Colli, Infinitely many coexisting strange attractors, Annales de l'Institut Henri Poincaré, Analyse non linéaire 15(5), 539–579 (1998). DOI: 10.1016/s0294-1449(98)80001-2. https://doi.org/10.1016/s0294-1449(98)80001-2
  8. Jacob Palis, A global view of dynamics and a conjecture on the denseness of finitude of attractors, Astérisque 261, exposé 86, 335-347 (2000). DOI: 10.24033/ast.478. https://doi.org/10.24033/ast.478