For the original Hénon family of planar diffeomorphisms, Newhouse (2004, Conjecture 1.3) conjectured that for each fixed parameter bb there is a residual (Baire-generic) set of parameters aa such that the map Ha,bH_{a,b} has no SRB (Sinai–Ruelle–Bowen) measure at all — no ergodic invariant probability measure with compact support and absolutely continuous conditional measures on unstable manifolds. For small b≠0b\neq 0, Kiriki–Li–Soma (2010) proved a local counterpart inside explicit parameter intervals JbJ_b: an open dense subset of JbJ_b carries no SRB measure on the homoclinic set of the fixed point (and a residual subset exhibits infinitely many sinks), yet a positive-Lebesgue-measure subset carries an SRB measure on a small Hénon-like strange attractor. The global statement — residual absence of every SRB measure for the whole map, for each bb — remains open. The problem asks for a proof or refutation of exactly this conjecture.

Background

The Hénon family is the two-parameter family of planar diffeomorphisms Ha,b(x,y)=(1+y−ax2,  bx)H_{a,b}(x,y) = (1 + y - ax^2,\; bx) introduced by Michel Hénon in 1976 as a minimal model of dissipative chaos; its Jacobian determinant is −b-b everywhere, so ∣b∣<1|b| < 1 gives area-decreasing (dissipative) maps, b=0b = 0 collapses the dynamics to the quadratic (logistic-type) interval map, and ∣b∣=1|b| = 1 gives area-preserving maps. Authors use equivalent normalizations, e.g. the smoothly conjugated φa,b(x,y)=(y,  a−bx+y2)\varphi_{a,b}(x,y) = (y,\; a - bx + y^2); the statements below are invariant under these changes of coordinates and reparametrization.

An SRB (Sinai–Ruelle–Bowen) measure is an invariant ergodic Borel probability measure with compact support whose conditional measures on unstable manifolds (the curvilinear directions along which nearby orbits separate exponentially) are absolutely continuous with respect to Lebesgue measure. Such measures are the mathematically rigorous version of the "physical" statistics observed along Lebesgue-typical orbits of chaotic attractors. A residual set is a countable intersection of open dense sets; by Baire's theorem such sets are dense, and holding on a residual set is the standard topological notion of "generic" — a much weaker requirement than holding on a positive-measure set.

For the measure-theoretic side, the work of Benedicks–Carleson (1991), Benedicks–Young (1993), and Mora–Viana (1993) shows that in Hénon and Hénon-like families with ∣b∣|b| small there is a positive Lebesgue measure set of parameters aa for which the map has an SRB measure supported on a strange attractor. On the topological side, a homoclinic tangency — a nontransverse intersection of the stable and unstable manifolds of a hyperbolic saddle — unfolds under parameter changes into extremely rich dynamics: Newhouse (1974/1979) proved that near such a tangency (associated with a dissipative saddle, i.e. one whose product of eigenvalue moduli is less than 1) there are open sets of diffeomorphisms in which a residual subset exhibits the Newhouse phenomenon: infinitely many coexisting attracting periodic orbits (sinks). Newhouse (2004, Theorem 1.4) strengthened the generic picture: for a residual set of CrC^r surface diffeomorphisms, no homoclinic set (the closure of the basic sets tied to a periodic orbit by transverse intersections) that contains a tangency and a dissipative periodic point carries an SRB measure.

Newhouse then posed his Conjecture 1.3, transferring this generic obstruction to the fixed Hénon family itself: for each parameter bb, there should be a residual set of parameters aa such that Ha,bH_{a,b} has no SRB measure. Kiriki–Li–Soma (2010) verified the mechanism inside the family: for every small b≠0b \neq 0 there is a closed aa-interval JbJ_b with persistent tangencies of the homoclinic set of a fixed point, in which an open dense subset of parameters has no SRB measure supported on that homoclinic set, a residual subset has infinitely many sinks, and a subset positive everywhere in Lebesgue measure has an SRB measure on a small Hénon-like strange attractor — the two worlds coexisting in the same interval. However, infinitely many sinks do not preclude SRB measures supported elsewhere, and excluding SRB measures on one homoclinic class is far from excluding every SRB measure of the whole map. The step from Newhouse's ambient Diffr\mathrm{Diff}^r perturbation arguments to the rigid two-parameter Hénon family, and from a single homoclinic class to the entire invariant-measure structure, is precisely what remains unresolved.

For the area-preserving sections ∣b∣=1|b| = 1, one consequence of the conjecture is the following: since the non-wandering set of Ha,bH_{a,b} is compact and, by the Pesin entropy formula and results of Ledrappier (and Ledrappier–Young), any ergodic component of normalized Lebesgue measure on an invariant set of finite area would be an SRB measure, the conjecture would imply that, for a residual set of parameters aa, Ha,bH_{a,b} has zero measure-theoretic entropy on any invariant set of finite area — equivalently, every ergodic component of normalized Lebesgue measure on such a set has zero entropy.

Problem Statement

Resolve Newhouse's Conjecture 1.3 (2004) for the original Hénon family. In Newhouse's normalization Ha,b(x,y)=(1+y−ax2,  bx)H_{a,b}(x,y) = (1 + y - ax^2,\; bx) (equivalently, up to the smooth reparametrization and conjugacy φa,b(x,y)=(y,  a−bx+y2)\varphi_{a,b}(x,y) = (y,\; a - bx + y^2) used by Kiriki–Li–Soma), the conjecture states:

for each fixed parameter b, there is a residual subset Rb of the a-parameter line such that Ha,b has no SRB measure for every a∈Rb.\text{for each fixed parameter } b, \text{ there is a residual subset } R_b \text{ of the } a\text{-parameter line such that } H_{a,b} \text{ has no SRB measure for every } a \in R_b.

Here an SRB measure is an invariant ergodic Borel probability measure with compact support whose conditional measures on unstable manifolds are absolutely continuous with respect to Lebesgue measure. The claim concerns nonexistence of any SRB measure of the entire map — not merely of measures supported on a prescribed homoclinic class — and it quantifies over every fixed parameter bb, with no restriction to small or dissipative values; the area-preserving sections ∣b∣=1|b| = 1 are therefore inside its scope. A complete answer is either a proof (construct the residual sets RbR_b and exclude every SRB measure, for each fixed bb) or a refutation (exhibit a fixed bb for which SRB-carrying parameters are so widespread that no such residual set exists, i.e. the set of aa for which Ha,bH_{a,b} has no SRB measure is not comeager, equivalently the SRB-carrying parameters are non-meager — they meet every residual set; mere density of SRB-carrying parameters would not suffice, since a dense meager set has a residual complement).

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