  The transfer operator (TO) formalism of the dynamical systems (DS) theory is
reformulated here in terms of the recently proposed supersymetric theory of
stochastic differential equations (SDE). It turns out that the stochastically
generalized TO (GTO) of the DS theory is the finite-time Fokker-Planck
evolution operator. As a result comes the supersymmetric trivialization of the
so-called sharp trace and sharp determinant of the GTO, with the former being
the Witten index, which is also the stochastic generalization of the Lefschetz
index so that it equals the Euler characteristic of the (closed) phase space
for any flow vector field, noise metric, and temperature. The enabled
possibility to apply the spectral theorems of the DS theory to the
Fokker-Planck operators allows to extend the previous picture of the
spontaneous topological supersymmetry (Q-symmetry) breaking onto the situations
with negative ground state's attenuation rate. The later signifies the
exponential growth of the number of periodic solutions/orbits in the large time
limit, which is the unique feature of chaotic behavior proving that the
spontaneous breakdown of Q-symmetry is indeed the field-theoretic definition
and stochastic generalization of the concept of deterministic chaos. In
addition, the previously proposed low-temperature classification of SDEs, i.e.,
thermodynamic equilibrium / noise-induced chaos ((anti)instanton condensation,
intermittent) / ordinary chaos (non-integrability of the flow vector field), is
complemented by the discussion of the high-temperature regime where the sharp
boundary between the noise-induced and ordinary chaotic phases must smear out
into a crossover, and at even higher temperatures the Q-symmetry is restored.
The Weyl quantization is discussed in the context of the Ito-Stratonovich
dilemma.
