Atlas: Dynamical system on Cantor set by Makoto Mori

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Dynamical Systems and Related Topics Workshop
March 21-24, 1998
University of Maryland
College Park, MD, USA

Organizers
Mike Boyle, Brian Hunt, Jim Yorke

Dynamical system on Cantor set
by
Makoto Mori
Dept.Math. College of Humanities and Sciences, Nihon university

Dynamical system on Cantor set

Makoto Mori

Feb. 22

One of the way to study the ergodic properties of a dynamical system generated by an expanding map from an interval into itself is to study the spectra of the corresponding Perron-Frobenius operator. We consider generating functions associated with the dynamical system. Then constructing a renewal equation, we define a `Fredholm Matrix' \Phi(z). We prove that the determinant det(I-\Phi(z)) plays a similar role as the Fredholm determinant in case of nuclear operators. Moreover, using the fact that the Fredholm Matrix \Phi(z) is essentially the structure matrix, we can show the Ruelle's zeta function satisfies \zeta(z)=(det(I-\Phi(z))^-1. Combining the above results, we also get the relations between the singularities of the zeta function and the spectra of the Perron-Frobenius operator.

Applying the above results to cantor set generated by an expanding map, we can determine not only the Hausdorff dimension of the cantor set but also the density function of the invariant measure absolutely continuous to the Hausdorff measure and study the ergodic properties of the dynamical system.

Date received: February 23, 1998