We construct a Nekhoroshev-like result of stability with sharp constants for the planar three body problem, both in the planetary and in the restricted circular case, by using the periodic averaging technique. Our constructions can be generalized to any near-integrable hamiltonian system whose unperturbed hamiltonian is quasi-convex. The dependence of the constants on the analyticity widths of the complex hamiltonian is carefully taken into account. This allows for a deep analytical understanding of the limits of such techniques in insuring Nekhoroshev stability for high magnitudes of the perturbation and suggests hints on how to overcome such obstructions in some cases. Finally, two examples with concrete values are considered, one for the planetary case and one for the restricted one.

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In this paper we prove the first result of Nekhoroshev stability for steep Hamiltonians in Hölder class. Our new approach combines the classical theory of normal forms in analytic category with an improved smoothing procedure to approximate an Hölder Hamiltonian with an analytic one. It is only for the sake of clarity that we consider the (difficult) case of Hölder perturbations of an analytic integrable Hamiltonian, but our method is flexible enough to work in many other functional classes, including the Gevrey one. The stability exponents can be taken to be $$(\ell -1)/(2n{\varvec{\alpha }}_1...{\varvec{\alpha }}_{n-2})+1/2$$ ( ℓ - 1 ) / ( 2 n α 1 . . . α n - 2 ) + 1 / 2 for the time of stability and $$1/(2n{\varvec{\alpha }}_1...{\varvec{\alpha }}_{n-1})$$ 1 / ( 2 n α 1 . . . α n - 1 ) for the radius of stability, n being the dimension, $$\ell >n+1$$ ℓ > n + 1 being the regularity and the $${\varvec{{\alpha }}}_i$$ α i ’s being the indices of steepness. Crucial to obtain the exponents above is a new non-standard estimate on the Fourier norm of the smoothed function. As a byproduct we improve the stability exponents in the $$C^k$$ C k class, with integer k.

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Consider an open, bounded set Ω ⊂ ℂ, a positive integer and a compact  ⊂ Ω of cardinality strictly greater than . We prove that, for any function which is holomorphic in Ω, and whose graph satisfies ( , ( )) = 0 for some polynomial ∈ ℂ[ , ] of degree at most (hence is an algebraic function), the quantity max Ω | |∕ max  | | is bounded by a constant that only depends on , Ω,  but not on (estimates of this kind are called Bernstein-Remez inequalities). This result has been demonstrated by Roytwarf and Yomdin in case  is a real interval, and later by Yomdin for a discrete set  of sufficiently high cardinality, by using arguments of real-algebraic and analytic geometry. Here we present and extend a proof due to Nekhoroshev on the existence of a uniform Bernstein-Remez inequality for algebraic functions, which relies on classical theorems of complex analysis. Nekhoroshev's work remained unstudied despite its important consequences in Hamiltonian dynamics and is here presented and extended in a self-contained and pedagogical way, while the original reasonings were rather sketchy.