On the distribution of mantissae in nonautonomous difference equations

Abstract

Mantissa distributions generated by dynamical processes continue to attract much interest. In this article, it is demonstrated that one-dimensional projections of (at least) almost all orbits of many multi-dimensional nonautonomous dynamical systems exhibit a mantissa distribution that is a convex combination of a trivial point mass and Benford's Law, i.e. the mantissa distribution of the non-trivial part of the orbit is asymptotically logarithmic, typically for all bases. Both linear and power-like systems are considered, and Benford behaviour is found to be ubiquitous for either class. The results unify previously known facts and extend them to the nonautonomous setting, with many of the conclusions being best possible in general.

Keywords:

• Nonautonomous dynamical system
• Benford's Law
• Uniform distribution mod 1 in ℝ ^d
• Exponential dichotomy
• Shadowing

Keywords:

• 11K06
• 37A45
• 37B55

Acknowledgements

The first author was partially supported by a humboldt Research Fellowship; he wishes to thank B. Martin for a helpful comment. The second author was supported by the emmy noether Program funded by the DFG.

Notes

^¶ Email:[email protected]

^§ When computed using the Euclidean norm, β _A >1 unless αA is unitary for some number α with |α| ≥ 1; the latter case clearly is irrelevant here.

Additional information

Notes on contributors

S. Siegmund

1 ¶ Email:[email protected]