Abstract
We introduce a new way of representing any real number in the unit interval as a sequence of positive integers. This representation has its physical motivation from considering a chaotic dynamical system. That is why we use the name “dynamical representation.” A reconstruction procedure for the real number from the given sequence of natural numbers is given. Important ergodic properties about the digits of representations, in particular those resembling Borel's theorem in binary expansion and Khintchine's theorem in continued fraction expansion, are derived. Finally, we invoke a generalized version of the Khintchine-Kolmogorov law of the iterated logarithm to obtain the sharpest possible estimate of the corresponding differences. A twofold universality for the distribution behavoir of symmetric dynamical representation is observed.
Published Version
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