Preface

Dynamics in all its variations is the study of change. In the usual physical

context, change takes place within time. The objects of geometry are static and

if there is any change, it is “in the eye of the beholder”. In fractal geometry this

point takes on meaning, particularly in the form of changing degree of magnifica-

tion and “zooming in” on an object. This suggests developing dynamical concepts

appropriate to this framework.

In these notes, based on a series of lectures delivered at Kent State University

in 2011, we show that ergodic theoretic concepts can be applied to the process of

changing magnification to give insight to phenomena peculiar to fractals. An im-

portant step is showing how fractal dimension can be interpreted in terms of ergodic

averages in an appropriate measure preserving system. The familiar phenomenon

of self similarity appears as the analogue of periodicity in classical dynamics. We

don’t pursue the full implications of recurrence in the geometric context, but some

examples of the related Ramsey type questions are considered.

The theory developed here and the major ideas originated in the papers [F] and

[F ]. It will develop that there is a close connection between dimension theory and

rates of growth of trees. This is exploited in [FW] where analogs of Szemer´edi’s

theorem are demonstrated in the context of trees.

I am indebted to Dmitry Ryabogin and Fedor Nazarov for transcribing the

lectures as well as for working out many details that were not provided in the

lectures as I presented them.

Hillel Furstenberg, January, 2014

Jerusalem, Israel

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