Digits & deviations of dynamical systems

Dynamical systems describe the evolution of objects in a space over time and may be used to model physical phenomena. Due to this, much research is dedicated to understanding the mathematics of dynamical systems. In this thesis, we quantify the unlikeliness of unlikely events defined via two different families of discrete-time dynamical systems.<br>The first family includes types of “non-autonomous number systems”. These are systems whose dynamics generate representations of numbers using digits in a way that is time-dependent, i.e. the description of the system’s evolution changes at each timestep. The events considered are sets of numbers whose digits appear in the corresponding representations with prescribed frequencies, and we quantify these sets’ sizes using the Hausdorff dimension, which measures fine geometric detail. The formula obtained generalises a known result for time-independent systems and exhibits the same ‘global lower bound’ phenomenon introduced there.<br>The second family includes dynamical systems with good “spectral properties”. From one such system, we define certain random variables, which can be thought of as outcomes of random events, e.g. getting tails from a coinflip. We consider the setting where S_n=X_1+X_2+⋯+X_n approaches an α-stable distribution (a generalisation of the standard Gaussian), and determine the large deviations of S_n, i.e. we quantify the rate at which the probability of S_n exceeding an appropriately large amount decays to zero. This generalises a well-known result in the independent setting to our dependent systems and is seemingly the first result of this type with conditions straightforward to verify for dynamical systems.<br>