Generic Birkhoff spectra

Buczolich, Zoltán and Maga, Balázs and Moore, Ryo (2020) Generic Birkhoff spectra. DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS SERIES A, 40 (12). pp. 6649-6679. ISSN 1078-0947

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Abstract

Suppose that Omega = {0, 1}(N) and sigma is the one-sided shift. The Birkhoff spectrum S-f(alpha) = dim(H) {omega is an element of Omega : lim(N ->infinity) 1/N Sigma(N)(n=1) f(sigma(n)omega) = alpha} where dim H is the Hausdorff dimension. It is well-known that the support of S-f(alpha) is a bounded and closed interval L-f = [alpha(f,min)* ,alpha(f,max)*]and S-f(alpha) on L-f is concave and upper semicontinuous. We are interested in possible shapes/properties of the spectrum, especially for generic/typical f is an element of C(Omega) in the sense of Baire category. For a dense set in C(Omega) the spectrum is not continuous on R, though for the generic f is an element of C(Omega) the spectrum is continuous on R, but has infinite one-sided derivatives at the endpoints of L-f. We give an example of a function which has continuous S-f on R, but with finite one-sided derivatives at the endpoints of L-f. The spectrum of this function can be as close as possible to a "minimal spectrum". We use that if two functions f and g are close in C(Omega) then S-f and S-g are close on L-f apart from neighborhoods of the endpoints.

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