Travel in an Infinite Desert

Abstract

A caravan traverses an infinite desert studded with oases. It can rest indefinitely at each oasis. Given the sequence of the oases’ locations, how does the number of the caravan’s itineraries grow with time? We show that the growth is exponential when the oasis sequence is asymptotically linear, and subexponential when the oasis sequence is superlinear. Moreover, the growth has to be superpolynomial, but can be barely so.

MSC::

• Primary 05A16
• Secondary 37B40
• 37C35

Acknowledgments

The authors wish to thank the anonymous referees for many helpful suggestions. This work was partially supported by grant number 426602 from the Simons Foundation to Michał Misiurewicz.

Additional information

Funding

This work was partially supported by grant number 426602 from the Simons Foundation to Michał Misiurewicz.

Notes on contributors

William Geller

WILLIAM GELLER lived in an oasis in the Negev desert for two years as a teenager. He received his undergraduate degree from Harvard University and his Ph.D. from the University of California at Berkeley. He held visiting positions at the Hebrew University of Jerusalem, the University of Maryland, the University of Warwick, and the Mathematical Sciences Research Institute before moving to Indiana University-Purdue University Indianapolis. In addition to topological aspects of dynamical systems, his interests include coarse geometry and game theory.

Michał Misiurewicz

MICHAŁ MISIUREWICZ received his Ph.D. in mathematics from the University of Warsaw, and worked there for the next 16 years. After coming to America, he held visiting positions at Northwestern University and Princeton University, and finally landed at Indiana University-Purdue University Indianapolis. After working there for 29 years, he just retired. He is a fellow of the American Mathematical Society and a foreign member of the Polish Academy of Sciences. His research interests are in dynamical systems, especially in one-dimensional dynamics and topological entropy.