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Register a new userM{o}bius disjointness for topological models of ergodic systems with discrete spectrum
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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.
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A vast class of exponential functions is showed to be deterministic. This class includes functions whose exponents are polynomial-like or piece-wise close to polynomials after differentiation. Many of these functions are indeed disjoint from the Mobius function. As a consequence, we show that Sarnaks Disjointness Conjecture for the Mobius function (from deterministic sequences) is equivalent to the disjointness in average over short intervals
In this paper it is shown that every non-periodic ergodic system has two topologically weakly mixing, fully supported models: one is non-minimal but has a dense set of minimal points; and the other one is proximal. Also for independent interests, for a given Kakutani-Rokhlin tower with relatively prime column heights, it is demonstrated how to get a new taller Kakutani-Rokhlin tower with same property, which can be used in Weisss proof of the Jewett-Kriegers theorem and the proofs of our theorems. Applications of the results are given.
We study the topological properties of Peierls transitions in a monovalent M{o}bius ladder. Along the transverse and longitudinal directions of the ladder, there exist plenty Peierls phases corresponding to various dimerization patterns. Resulted from a special modulation, namely, staggered modulation along the longitudinal direction, the ladder system in the insulator phase behaves as a ``topological insulator, which possesses charged solitons as the gapless edge states existing in the gap. Such solitary states promise the dispersionless propagation along the longitudinal direction of the ladder system. Intrinsically, these non-trivial edges states originates from the Peierls phases boundary, which arises from the non-trivial $mathbb{Z}^{2}$ topological configuration.
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.
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 $Mtoinfty$, $Htoinfty$, $H/Mto 0$.
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