On Spectral Properties of Skew Products over Irrational Rotations

Abstract Entropy dimension is an entropy-type quantity which takes values in $[0,1]$ and classifies different levels of intermediate growth rate of complexity for dynamical systems. In this paper, we consider the complexity of skew products of irrational rotations with Bernoulli systems, which can be viewed as deterministic walks in random sceneries, and show that this class of models can have any given entropy dimension by choosing suitable rotations for the base system.

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On torus homeomorphisms semiconjugate to irrational rotations

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2014.23 ◽

2014 ◽

Vol 35 (7) ◽

pp. 2114-2137 ◽

Cited By ~ 8

Author(s):

T. JÄGER ◽

A. PASSEGGI

Keyword(s):

Simple Proof ◽

Irrational Rotation ◽

Rotation Vector ◽

Topological Properties ◽

Empty Interior ◽

Irrational Rotations ◽

Skew Products ◽

Invariant Foliation ◽

Special Case

In the context of the Franks–Misiurewicz conjecture, we study homeomorphisms of the two-torus semiconjugate to an irrational rotation of the circle. As a special case, this conjecture asserts uniqueness of the rotation vector in this class of systems. We first characterize these maps by the existence of an invariant ‘foliation’ by essential annular continua (essential subcontinua of the torus whose complement is an open annulus) which are permuted with irrational combinatorics. This result places the considered class close to skew products over irrational rotations. Generalizing a well-known result of Herman on forced circle homeomorphisms, we provide a criterion, in terms of topological properties of the annular continua, for the uniqueness of the rotation vector. As a byproduct, we obtain a simple proof for the uniqueness of the rotation vector on decomposable invariant annular continua with empty interior. In addition, we collect a number of observations on the topology and rotation intervals of invariant annular continua with empty interior.

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On Weyl sums and skew products over irrational rotations

Theoretical Computer Science ◽

10.1016/0304-3975(89)90043-1 ◽

1989 ◽

Vol 65 (2) ◽

pp. 189-196 ◽

Cited By ~ 7

Author(s):

P. Hellekalek ◽

G. Larcher

Keyword(s):

Irrational Rotations ◽

Skew Products

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Model of multicomponent narrow-band periodical non-stationary random signal

Information extraction and processing ◽

10.15407/vidbir2020.48.017 ◽

2020 ◽

Vol 2020 (48) ◽

pp. 17-24

Author(s):

I.M. Javorskyj ◽

◽

R.M. Yuzefovych ◽

P.R. Kurapov ◽

◽

...

Keyword(s):

Time Series ◽

Real Time ◽

Hilbert Transform ◽

Spectral Properties ◽

Narrow Band ◽

Analytic Signal ◽

Random Signal ◽

Hilbert Transformation ◽

The Hilbert Transform

The correlation and spectral properties of a multicomponent narrowband periodical non-stationary random signal (PNRS) and its Hilbert transformation are considered. It is shown that multicomponent narrowband PNRS differ from the monocomponent signal. This difference is caused by correlation of the quadratures for the different carrier harmonics. Such features of the analytic signal must be taken into account when we use the Hilbert transform for the analysis of real time series.

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