We review the convergence of chaotic integrals computed by Monte Carlo simulation, the trace method, dynamical zeta function, and Fredholm determinant on a simple one-dimensional example: the parabola repeller. There is a dramatic difference in convergence between these approaches. The convergence of the Monte Carlo method follows an inverse power law, whereas the trace method and dynamical zeta function converge exponentially, and the Fredholm determinant converges faster than any exponential.
REFERENCES
1.
P.
Cvitanović
, “Invariant measurements of strange sets in terms of cycles
,” Phys. Rev. Lett.
61
, 2729
–2732
(1988
).2.
L. P.
Kadanoff
and C.
Tang
, “Escape from strange repellers
,” Proc. Natl. Acad. Sci., USA
81
, 1276
–1279
(1984
).3.
T.
Tel
, “Escape rate from strange sets as an eigenvalue
,” Phys. Rev. A
36
, 1502
–1505
(1987
).4.
A. Grothendieck, “Produits tensoriels topologiques et éspaces nucléaires,” in Memoirs of the American Mathematical Society (American Mathematical Society, Providence, 1955), Vol. 16.
5.
J. Milnor and W. Thurston, “On iterated maps of the interval,” in Dynamical Systems: Proceedings of the Special Year Held at the University of Maryland, College Park, 1986–87, Lecture Notes in Mathematics (Springer-Verlag, Berlin, 1988), Vol. 1342, pp. 465–563.
6.
K. T. Hansen, “Symbolic dynamics in chaotic systems,” Ph.D. thesis, University of Oslo, 1993.
7.
R.
Artuso
, E.
Aurell
, and P.
Cvitanović
, “Recycling of strange sets: I. Cycle expansions
, ” Nonlinearity
3
, 325
–359
(1990
).8.
P.
Cvitanović
, “Dynamical averaging in terms of periodic orbits
,” Physica D
83
, 109
–123
(1995
).9.
H. H.
Rugh
, “The correlation spectrum for hyperbolic analytic maps
,” Nonlinearity
5
, 1237
–1263
(1992
).10.
R.
Stoop
and J.
Parisi
“Evaluation of probabilistic and dynamic invariants from finite symbolic substrings—comparison between 2 approaches
,” Physica D
58
, 325
–330
(1992
).11.
G.
Keller
, “On the rate of convergence to equilibrium in one-dimensional systems
,” Commun. Math. Phys.
96
, 181
–193
(1984
).12.
E.
Aurell
, “Convergence of dynamic zeta-functions
,” J. Stat. Phys.
58
, 967
–995
(1990
).13.
P.
Cvitanović
, “Periodic orbits as the skeleton of classical and quantum chaos
,” Physica D
51
, 138
–151
(1991
).14.
R.
Mainieri
, “Zeta function for the Lyapunov exponent of a product of random matrices
,” Phys. Rev. Lett.
68
, 1965
–1968
(1992
).15.
R.
Mainieri
, “Cycle expansions with pruned orbits have branch-points
,” Physica D
83
, 206
–215
(1995
).
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© 1997 American Institute of Physics.
1997
American Institute of Physics
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