seminar: New explicit conditions for steepness in the space of five-jets of smooth functions
speaker: Santiago Barbieri (Università di Padova)abstract: The notion of steepness for a given function was firstly introduced by N.N. Nekhoroshev in the Seventies. A function which is real-analytic around a compact set of $\mathbb{R}n$ is steep if and only if it has no critical points and its restriction to any affine hyperplane of $\mathbb{R}n$ admits only isolated critical points (Niederman, 2006). Such property is related to the stability of finite-dimensional hamiltonian systems which are close to integrable, i.e. whose hamiltonian function is the sum of an integrable part and a small perturbation. By making use of canonical action-angle coordinates to describe such system, it is well known that the actions stay constant under the integrable flow, whereas diffusion is possible when a perturbation is added, provided the system has three or more degrees of freedom. A celebrated result by Nekhoroshev (1977) states that, nevertheless, the action variation is small over a time which is exponential in the inverse of the size of the perturbation, provided that the integrable part of the hamiltonian is steep. In this light, understanding wether a given function is steep or not turns out to be a crucial issue. Unfortunately, the simple definition of steepness is not easy to verify directly. However, in his work of 1979, Nekhoroshev provided the way to explicitly construct a semi-algebraic set in the space of r-jets of functions (that is, the space of the Taylor polynomials calculated at a certain point x, up to a fixed order r\geq 2) whose closure contains the jets of all non-steep functions with non-zero gradient at x. Indeed, a function whose r-jet at x is contained in the complementary of the closure of such semi-algebraic set is steep at x. The explicit computations involved in this construction have been explicitly carried out by Nekhoroshev (1977) for r=2,3 and by Schirinzi and Guzzo (2013) for r=4. In this seminar, I will show the construction for r=5 and, therefore, provide new explicit algebraic conditions for steepness on the coefficients of the Taylor polynomial of order five of a given smooth function. Besides loosening the sufficient conditions for a function to be steep, this works also allows for a deeper insight in the construction of sufficient conditions at any order r of the jet and this, in turn, leads to the formulation of interesting conjectures
timetable:
Thu 28 Mar, 14:30 - 15:30, Sala Conferenze Centro De Giorgi
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