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Register a new userOn spectral disjointness of powers for rank-one transformations and Mobius orthogonality
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El Houcein El Abdalaoui
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We study the spectral disjointness of the powers of a rank-one transformation. For a large class of rank-one constructions, including those for which the cutting and stacking parameters are bounded, and other examples such as rigid generalized Chacons maps and Katoks map, we prove that different positive powers of the transformation are pairwise spectrally disjoint on the continuous part of the spectrum. Our proof involves the existence, in the weak closure of {U_T^k: k in Z}, of sufficiently many analytic functions of the operator U_T. Then we apply these disjointness results to prove Sarnaks conjecture for the (possibly non-uniquely ergodic) symbolic models associated to these rank-one constructions: All sequences realized in these models are orthogonal to the Mobius function.
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In this paper we give explicit characterizations, based on the cutting and spacer parameters, of (a) which rank-one transformations factor onto a given finite cyclic permutation, (b) which rank-one transformations factor onto a given odometer, and (c) which rank-one transformations are isomorphic to a given odometer. These naturally yield characterizations of (d) which rank-one transformations factor onto some (unspecified) finite cyclic permutation, (d) which rank-one transformations are totally ergodic, (e) which rank-one transformations factor onto some (unspecified) odometer, and (f) which rank-one transformations are isomorphic to some (unspecified) odometer.
We provide a criterion for a point satisfying the required disjointness condition in Sarnaks Mobius Disjointness Conjecture. As a direct application, we have that the conjecture holds for any topological model of an ergodic system with discrete spectrum.
We show that joinings of higher rank torus actions on S-arithmetic quotients of semi-simple or perfect algebraic groups must be algebraic.
We show that Sarnaks conjecture on Mobius disjointness holds in every uniquely ergodic modelof a quasi-discrete spectrum automorphism. A consequence of this result is that, for each non constant polynomial $PinR[x]$ with irrational leading coefficient and for each multiplicative function $bnu:NtoC$, $|bnu|leq1$, we have[ frac{1}{M} sum_{Mle mtextless{}2M} frac{1}{H} left| sum_{mle n textless{} m+H} e^{2pi iP(n)}bnu(n) right|longrightarrow 0 ] as $Mtoinfty$, $Htoinfty$, $H/Mto 0$.
Let $R$ be a ring of characteristic $0$ with field of fractions $K$, and let $mge2$. The Bottcher coordinate of a power series $varphi(x)in x^m + x^{m+1}R[![x]!]$ is the unique power series $f_varphi(x)in x+x^2K[![x]!]$ satisfying $varphicirc f_varphi(x) = f_varphi(x^m)$. In this paper we study the integrality properties of the coefficients of $f_varphi(x)$, partly for their intrinsic interest and partly for potential applications to $p$-adic dynamics. Results include: (1) If $p$ is prime and $R=mathbb Z_p$ and $varphi(x)in x^p + px^{p+1}R[![x]!]$, then $f_varphi(x)in R[![x]!]$. (2) If $varphi(x)in x^m + mx^{m+1}R[![x]!]$, then $f_varphi(x)=xsum_{k=0}^infty a_kx^k/k!$ with all $a_kin R$. (3) In (2), if $m=p^2$, then $a_kequiv-1pmod{p}$ for all $k$ that are powers of $p$.
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