| He introduced ergodic methods into number theory in his first work. |
| Armed with this understanding of the definition of ergodicity, we can now discuss some important properties of ergodic systems. |
| While in the Harvard College Observatory he worked on many mathematical and astronomical subjects including topology and ergodic theory. |
| Doeblin also contributed to the theory of random chains with complete connection, some of which was used in a paper by him on ergodic properties of continued fractions. |
| For the quadrant version, we prove the existence of temporally ergodic evolutions and of spatially ergodic ones. |
| This duality structure, interesting by itsetf, is also instrumental in proving some of the ergodic results. |
