Currently I’m interested in a couple of fields, namely dynamical systems, stochastic processes, and additive combinatorics. I was wondering if it’s feasible to keep pursuing all 3, and whether I can expect any overlap between the fields. I have not really narrowed down my focus too much yet since I’m just starting out, but to give an idea of the specific areas of interest, I’m currently using, or planning to use following books:
Katok & Hasselblatt - An introduction to the modern theory of dynamical systems
Einsiedler & Ward - Ergodic Theory with a view towards number theory
Eisner et. al. - Operator theoretic aspects of Ergodic theory
Furstenberg - Recurrence in Combinatorics and Number Theory (Planned)
- Le Gall - Brownian motion, Martingales and Stochastic Calculus
- Tao & Vu - Additive combinatorics
For reference I have a little under a year and a half before I start my PhD. Is there any overlap between these fields where I can combine some of my interests? Or should I specialise at some point after going through these general books?