Unit:
Κατεύθυνση Θεωρητικά ΜαθηματικάLibrary of the School of Science
Original Title:
Μια «version» του Θεωρήματος των Wiener-Wintner
Translated title:
A version of Wiener’s-Wintner’s Theorem
Summary:
This thesis presents a version of Wiener’s-Wintner’s Theorem (like the one that B. Weis gave in his paper-see bibliograrhy of the thesis):
“For each measure preserving transformatioin "T" of the probability space (X,B,m) and for each f∈L^1 (X,B,m), there exist a set Ν of measure zero, with respect to m, such that for each x∈X∖N, and for all the a∈S^1, the averages
1/n ∑_(k=0)^(n-1)▒〖a^k f(T^k x) converge^' s a.e.with respect to m〗.
Version of the Theorem: “ If no power of e^2πia is an eigenvalue of the ergodic transformation (X,Τ,μ) then the unique invariant measure on (X×Τ^1,Τ×R_α ) is μ×(Lebesgue measure), and in that case for every generic point x_0 and every continuous function f we have lim┬(n→∞)〖1/n ∑_(k=0)^(n-1)▒〖e^2πika f(T^k x_0 ) 〗〗=0 .”
The proof of the version of theorem is like the one that Furstenberg gave and saws why in a such general dynamical system like the (X,T,m), the asymptotic behavior almost all (with respect to m) the point’s of X combined (with the direct way that Wiener’s-Wintner’s Theorem gives) with a very specific situation, this of rotation’s of the circle S^1.
Keywords:
generic point, ergodic transformation, Wiener-Wintner Theorem
File:
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Kr_preamble.pdf
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