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Register a new userStrictly ergodic distal models and a new approach to the Host-Kra factors
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Cocycles are a key object in Antol{i}n Camarena and Szegedys (topological) theory of nilspaces. We introduce measurable counterparts, named nilcycles, enabling us to give conditions which guarantee that an ergodic group extension of a strictly ergodic distal system admits a strictly ergodic distal topological model, revisiting a problem studied by Lindenstrauss. In particular we show that if the base space is a dynamical nilspace then a dynamical nilspace topological model may be chosen for the extension. This approach combined with a structure theorem of Gutman, Manners and Varj{u} applied to the ergodic group extensions between successive Host-Kra characteristic factors gives a new proof that these factors are inverse limit of nilsystems.
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The classical theorem of Jewett and Krieger gives a strictly ergodic model for any ergodic measure preserving system. An extension of this result for non-ergodic systems was given many years ago by George Hansel. He constructed, for any measure preserving system, a strictly uniform model, i.e. a compact space which admits an upper semicontinuous decomposition into strictly ergodic models of the ergodic components of the measure. In this note we give a new proof of a stronger result by adding the condition of purity, which controls the set of ergodic measures that appear in the strictly uniform model.
A recent result of Downarowicz and Serafin (DS) shows that there exist positive entropy subshifts satisfying the assertion of Sarnaks conjecture. More precisely, it is proved that if $y=(y_n)_{nge 1}$ is a bounded sequence with zero average along every infinite arithmetic progression (the Mobius function is an example of such a sq $y$) then for every $Nge 2$ there exists a subshift $Sigma$ over $N$ symbols, with entropy arbitrarily close to $log N$, uncorrelated to $y$. In the present note, we improve the result of (DS). First of all, we observe that the uncorrelation obtained in (DS) is emph{uniform}, i.e., for any continuous function $f:Sigmato {mathbb R}$ and every $epsilon>0$ there exists $n_0$ such that for any $nge n_0$ and any $xinSigma$ we have $$ left|frac1nsum_{i=1}^{n}f(T^ix),y_iright|<epsilon. $$ More importantly, by a fine-tuned modification of the construction from (DS) we create a emph{strictly ergodic} subshift, with all the desired properties of the example in (DS) (uniformly uncorrelated to $y$ and with entropy arbitrarily close to $log N$). The question about these two additional properties (uniformity of uncorrelation and strict ergodicity) has been posed by Mariusz Lemanczyk in the context of the so-called strong MOMO (Mobius Orthogonality on Moving Orbits) property. Our result shows, among other things, that strong MOMO is essentially stronger than uniform uncorrelation, even for strictly ergodic systems.
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.
Let $Gamma$ be a lattice in ${rm SL}(n, mathbb R)$ with $ngeq 3$ and $mathcal S$ be a closed surface. Then $Gamma$ has no distal minimal action on $mathcal S$.
A new approach is given for the implementation of boundary conditions used in solving the Mukhanov-Sasaki equation in the context of inflation. The familiar quantization procedure is reviewed, along with a discussion of where one might expect deviations from the standard approach to arise. The proposed method introduces a (model dependent) fitting function for the z/z and a/a terms in the Mukhanov-Sasaki equation for scalar and tensor modes, as well as imposes the boundary conditions at a finite conformal time. As an example, we employ a fitting function, and compute the spectral index, along with its running, for a specific inflationary model which possesses background equations that are analytically solvable. The observational upper bound on the tensor to scalar ratio is used to constrain the parameters of the boundary conditions in the tensor sector as well. An overview on the generalization of this method is also discussed.
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