Tokyo, Japan

Phase space of multi-dimensional Hamiltonian systems is generally composed of infinitely many invariant components. Chaotic trajectories have the largest dimension as an invariant set, while the periodic orbits have the lowest. Coexistence of qualitatively different ergodic components, which are usually intermingled in a self-similar way in the phase space, characterizes a generic situation which is so complicated that our understanding is far from accomplished. The orbits in classical mechanics are always confined on the corresponding invariant set by definition, in particular, except in case of ideal chaotic systems, there are orbits with positive measure that move only on the limited subspace whose dimension is less than that of full phase space.

On the other hand, the wavepacket of quantum mechanics
is not forced to stay on a certain limited classical manifold, but
spreads over or shares different invariant subsets simultaneously.
The spreading is a consequence of the wave effect which is the most
marked difference between classical and quantum mechanics.
There is not any obstacle in principle preventing the transition between
arbitrary
two points in the phase space and the quantum wavepacket can penetrate
into any kinds of barriers. Such a classically forbidden process
does not have classical counterparts.
The penetration into the energy barrier is especially called *tunneling*, which is
understood as the most typical quantum effect and plays important roles in
many physical and chemical phenomena. The existence of chaos in
the phase space crucially affects the nature of tunneling
(Shudo & Ikeda 1995, 1998).

In this lecture, after introducing recent developments of the theory of
multi-dimensional complex dynamical systems, with some technical tools
necessary to construct the theory
(Bedford & Smillie 1991a, 1991b 1992), we will give numerical and
mathematical
evidences which show that the orbits on the Julia set in the complex phase
space
can just be regarded as the *classical counterparts* of the quantum
wave effects (Shudo, Ishii & Ikeda 2001).
More precisely, arbitrary two regions in the phase space are necessarily
connected via the orbits on the Julia set *even in the mixed system. *
This remarkable property, which is absent in the real classical dynamics,
comes from the transitivity of the Julia set, which has rigorously been
proved for
the complex Hénon map, and conjectured for the standard and semi-standard
map.
The measure whose support gives the Julia set is a unique ergodic measure,
and
unstable saddles are dense on it. Our arguments are based on the complex
semiclassical description of quantum forbidden processes.

**References**

Shudo A and Ikeda K S 1995 *Phys. Rev. Lett. ***74** 682.

Shudo A and Ikeda K S 1998 *Physica D* **115** 234.

Bedford E and Smillie J 1991a *Invent. Math. ***103** 69;

Bedford E and Smillie J 1991b *J. Amer. Math. Soc. ***4** 657.

Bedford E and Smillie J 1992 *Math. Ann.* **294** 395.

Shudo A, Ishii Y and Ikeda K S 2001 *J. Phys. A* in press.