Charles Pugh

The C 1 Closing Lemma, including Hamiltonians

An Axiom of Lift for classes of dynamical systems is formulated. It is shown to imply the Closing Lemma. The Lift Axiom is then verified for dynamical systems ranging from C 1 diffeomorphisms to C 1 Hamiltonian vector fields.

Ergodicity of Anosov actions

Definition [5]. Let G be a Lie group acting differentiably on M, A: G--,Diff(M) where M is a compact smooth manifold. We assume that the orbits of G define a differentiable foliation o~, which is the case for instance if the G action is locally free (every isotropy group is discrete), The action is called Anosov if there exists an Anosov e l e m e n t a n element g~G such that A ( g ) = f is hyperbolic at ~ [5] and (1) the G action is locally free, or (2) G is connected and g is central in G. We recall that A(g) = f is hyperbolic at ~ means that T f : TM ~ TM leaves invariant a splitting

Keeping track of limit cycles

This article uses analytic geometry methods to bound the number of limit cycles of bounded period for a planar polynomial vector fields. It introduces the notion of limit periodic sets and shows that the Hilberts 16th problem can be reduced to the existence of a uniform bound on its cyclicity. It also shows the existence of a universal bound on the number of limit cycles in the perturbation of a linear focus as a consequence of the Noetherian property of the Bautin ideal.

Stable ergodicity and Anosov flows

In this note we prove that if M is a 3-manifold and ϕt:M→M is a C2, volume-preserving Anosov flow, then the time-1 map ϕ1 is stably ergodic if and only if ϕt is not a suspension of an Anosov diffeomorphism.

Critical sets in 3-space

Given a non-empty compact set C ⊂R3, is C the set of critical points for some smooth proper functionf :R3 →R+? In this paper we prove that the answer is “yes” for Antoine’s Necklace and most but not all tame links.

The C1 Connecting Lemma

Given a smooth flow and two orbits such that theω-limit set of the first meets theα-limit set of the second, can the flow be C1-perturbed to make the orbits coincide? The answer is shown to be “not always.”

On the Validity of the Conjugate Pairing Rule for Lyapunov Exponents

For Hamiltonian systems subject to an external potential which in the presence of a thermostat will reach a nonequilibrium stationary state Dettmann and Morriss proved a strong conjugate pairing rule (SCPR) for pairs of Lyapunov exponents in the case of isokinetic (IK) stationary states which have a given kinetic energy. This SCPR holds for all initial phases of the system, all times t, and all numbers of particles N. This proof was generalized by Wojtkowski and Liverani to include hard interparticle potentials. A geometrical reformulation of those results is presented. The present paper proves numerically, using periodic orbits for the Lorentz gas, that SCPR cannot hold for isoenergetic (IE) stationary states which have a given total internal energy. In that case strong evidence is obtained for CPR to hold for large N and t, where it can be conjectured that the larger N, the smaller t will be. This suffices for statistical mechanics.

On Focal stability in dimension two

In Kupka et al. 2006 appears the Focal Stability Conjecture: the focal decomposition of the generic Riemann structure on a manifold M is stable under perturbations of the Riemann structure. In this paper, we prove the conjecture when M has dimension two, and there are no conjugate points.

Focal stability of Riemann metrics

Abstract Let M be a complete Riemann manifold with dimension m and metric g. For p, q ∈ M and ℓ > 0, let the index I (g, p, q, ℓ) be the number of g-geodesics of length ℓ that join p to q. The following generic bounds for this index are the main results we present here. We denote by ℛ the space of complete Riemann metrics on M. (a) For each p ∈ M, there is a residual 𝒢 (p) ⊂ ℛ such that for all g ∈ 𝒢(p) (b) If M is compact, there is a residual 𝒢 ⊂ ℛ such that for all g ∈ 𝒢 These finiteness results are part of our study of the focal decomposition—i.e., the partition Stability of this focal deomposition (as g varies) has a natural meaning, in analogy with structural stability in the theory of dynamical systems, and here we begin an investigation in that direction. Our methods involve the multi-transversality theory of J. Mather and the Bumpy Metric Theorem of R. Abraham, as proved by D. Anosov.

Partial Differentiability of Invariant Splittings

A key feature of a general nonlinear partially hyperbolic dynamical system is the absence of differentiability of its invariant splitting. In this paper, we show that often partial derivatives of the splitting exist and the splitting depends smoothly on the dynamical system itself.