One-dimensional dynamics owns many deep results and avenues of active mathematical research. Numerous inroads to this research exist for the advanced undergraduate or beginning graduate student. It is precisely these students whom we target. Several glimpses into one-dimensional dynamics are provided with the hope that the results presented illuminate the beauty and excitement of the field. Many topics covered appear nowhere else in “textbook format,” some are mini new research topics in themselves, and for nearly all topics we try to provide novel connections with other research areas both inside and outside the text. Among these topics are kneading theory and Hofbauer towers; detailed structure of ω-limit sets; topological entropy; lapnumbers and Markov extensions; the 2∞ map (Feigenbaum- Coullet-Tresser), interplay amongst continued fractions, adding machines, circle maps, and unimodal maps; irrational rotations as factors of unimodal maps; connections between β-transformations and unimodal maps; Ledrappier's three-dot example; and itineraries for complex quadratic maps and Hubbard trees. The flavor is largely combinatoric, symbolic, and topological. The material presented is notmeant to be approached in a linear fashion. Rather, we strongly encourage readers to pick and choose topics of interest. Trail routes (other than n ↦ n + 1) are indicated in Figure 1; more explicit information is provided at the beginning of each chapter. Suggested uses for the text include: dynamics courses, master theses, reading courses, research experiences for undergraduates (REUs), seminars, senior projects, and summer courses.

As mentioned, the topics covered are notthe typical topics seen in undergraduate/graduate dynamics texts. Rather, the material is a filtering from the research literature of currently active topics that can be made accessible to the targeted student audience.