Statistics of a Family of Piecewise Linear Maps
J. J. P. Veerman, P. J. Oberly, L. S. Fox

TL;DR
This paper investigates the statistical behavior of a family of piecewise linear maps, revealing that their deviations do not follow standard distributions like the Q-Gaussian, unlike the special case of rotations.
Contribution
It provides the first analytical evidence that deviations in these maps do not conform to Q-Gaussian or standard distributions, expanding understanding of their statistical properties.
Findings
Deviations do not follow Q-Gaussian distributions.
The limit distribution is non-standard and not smooth.
The case of rotation maps is a unique exception.
Abstract
We study statistical properties of the truncated flat spot map . In particular, we investigate whether for large , the deviations upon rescaling satisfy a -Gaussian distribution if and are both independently and uniformly distributed on the unit circle. This was motivated by the fact that if is the rotation by , then it has been shown that in this case the rescaled deviations are distributed as a -Gaussian with (a Cauchy distribution). This is the only case where a non-trivial (i.e. ) -Gaussian has been analytically established in a conservative dynamical system. In this note, however, we prove that for the family considered here, converges to a random variable with a curious distribution which is clearly not a -Gaussian or any other standard smooth distribution.
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Taxonomy
TopicsMathematical Dynamics and Fractals · Stochastic processes and statistical mechanics · Morphological variations and asymmetry
