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Session 16. Ergodic Theory and Dynamical Systems
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Correlation with strictly ergodic sequences |
Tomasz Downarowicz, IMPAN and Wroclaw Univeristy of Technology, Poland
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The talk is based on the joint work with Jacek Serafin
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A symbolic sequence is called strictly ergodic if so is its
shift orbit closure. Two symbolic sequences \(x=(x_n), y=(y_n)\)
(over an alphabet consisting of complex numbers) are
uncorrelated if the averages \(\frac 1n\sum_{i=1}^n
x_n\overline{y_n}\) tend to zero. We are interested in determining
which sequences are correlated to at least one strictly ergodic
sequence. The family includes all sequences that are generic for
ergodic measures (this follows easily from a theorem of Benjy Weiss
[1, Theorem 4,4], but not only such. In fact, we believe
that the sequences that do not have that property, is quite
exceptional. The motivation for this study is the question whether
the Möbius function is among the exceptional sequences, posed
(most likely) by M. Boshernitzan.
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References- B. Weiss, Single Orbit Dynamics , CBMS
Regional Conference Series in Mathematics, 95., American
Mathematical Society, Providence, RI, 2000.
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