Hyperbolic sets for twist maps
نویسنده
چکیده
An example is given of an area-preserving monotone twist map such that a uniformly hyperbolic structure exists on the closure of its Birkhoff maximizing orbits. This note provides a rigorous example of an area preserving monotone twist map / with the property that Df\ s has a uniformly hyperbolic structure, where B denotes the closure of the Birkhoff maximizing orbits. As shown by Mather [8] and by Aubry, La Daeron, and Andre [3], the set B associated with / contains invariant Cantor sets of all possible rotation numbers. A result which would imply the hyperbolicity of these invariant Cantor sets was announced by Aubry in [1]. The heuristic justification given there is discussed further in [2]. Nevertheless, Katok raises the hyperbolicity question again in [4] and [5]. The construction below gives a rigorous answer based on an estimate first due to Aubry. Another proof that hyperbolic Cantor sets A can exist in B was obtained independently by Michel Herman. Consider the 'standard' one parameter family of area preserving monotone twist maps of the cylinder T xR. One lift to R of the map in this family corresponding to the parameter k has the form The function I k k \ f(x, y) = ( x + y — sin 2irx, y -— sin 2TTX I. h(x,x') = -\(x-x') ^COS2TTX AIT generates / in the sense that f(x, y) = (x, y') if and only if y = —(x,x') and y' = — (x,x'). aX aX Thus, given a sequence {*„}, there exists a sequence {>>„} such that (xn, yn)=f(x0, y0) if and only if {*„} satisfies Ah Ah — (xn_,,xn)+ — (xm xn+1) = 0, (1) oX oX at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0143385700002996 Downloaded from https://www.cambridge.org/core. IP address: 54.191.40.80, on 08 Sep 2017 at 18:26:54, subject to the Cambridge Core terms of use, available
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تاریخ انتشار 2007