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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