Search

The least squares linear filter, also called the Wiener filter, is a popular tool to predict the next element(s) of time series by linear combination of time-delayed observations. We consider observation sequences of deterministic dynamics, and ask: Which pairs of observation function and dynamics are predictable? If one allows for nonlinear mappings of time-delayed observations, then Takens’ well-known theorem implies that a set of pairs, large in a specific topological sense, exists for which an exact prediction is possible. We show that a similar statement applies for the linear least squares filter in the infinite-delay limit, by considering the forecast problem for invertible measure-preserving maps and the Koopman operator on square-integrable functions. [ABSTRACT FROM AUTHOR]

Measure-theoretic slow entropy is a more refined invariant than the classical measure-theoretic entropy to characterize the complexity of dynamical systems with subexponential growth rates of distinguishable orbit types. In this paper we prove flexibility results for the values of upper and lower polynomial slow entropy of rigid transformations as well as maps admitting a good cyclic approximation. Moreover, we show that there cannot exist a general upper bound on the lower measure-theoretic slow entropy for systems of finite rank. [ABSTRACT FROM AUTHOR]

Suppose (f,\mathcal {X},\mu) is a measure preserving dynamical system and \phi \colon \mathcal {X}\to \mathbb {R} a measurable function. Consider the maximum process M_n≔\max \{X_1,\ldots,X_n\}, where X_i=\phi \circ f^{i-1} is a time series of observations on the system. Suppose that (u_n) is a non-decreasing sequence of real numbers, such that \mu (X_1>u_n)\to 0. For certain dynamical systems, we obtain a zero–one measure dichotomy for \mu (M_n\leq u_n\,\text {i.o.}) depending on the sequence u_n. Specific examples are piecewise expanding interval maps including the Gauß map. For the broader class of non-uniformly hyperbolic dynamical systems, we make significant improvements on existing literature for characterising the sequences u_n. Our results on the permitted sequences u_n are commensurate with the optimal sequences (and series criteria) obtained by Klass (1985) for i.i.d. processes. Moreover, we also develop new series criteria on the permitted sequences in the case where the i.i.d. theory breaks down. Our analysis has strong connections to specific problems in eventual always hitting time statistics and extreme value theory. [ABSTRACT FROM AUTHOR]

We study shrinking target problems and the set of eventually always hitting points. These are the points whose first n iterates will never have empty intersection with the nth target for sufficiently large n. We derive necessary and sufficient conditions on the shrinking rate of the targets for to be of full or zero measure especially for some interval maps including the doubling map, some quadratic maps and the Manneville–Pomeau map. We also obtain results for the Gauß map and correspondingly for the maximal digits in continued fraction expansions. In the case of -transformations we also compute the packing dimension of complementing already known results on the Hausdorff dimension of. [ABSTRACT FROM AUTHOR]

In continuation of Matsumoto’s paper (Nonlinearity 25:1495-1511, 2012) we show that various subspaces are C∞-dense in the space of orientation-preserving C∞-diffeomorphisms of the circle with rotation number α, where α∈S1 is any prescribed Liouville number. In particular, for every odometer O of product type we prove the denseness of the subspace of diffeomorphisms which are orbit-equivalent to O. [ABSTRACT FROM AUTHOR]

In the case of the disc D 2 , the annulus A = S 1 × [ 0 , 1 ] and the torus T 2 we will show that if a sequence of natural numbers satisfies a certain growth rate, then there is a weak mixing diffeomorphism that is uniformly rigid with respect to that sequence. The proof is based on a quantitative version of the Anosov–Katok-method with explicitly defined conjugation maps and the constructions are done in the C ∞ -topology. Beyond that we can deduce a similar result in the real-analytic topology in the case of T 2 . [ABSTRACT FROM AUTHOR]