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Abstract:
In many situations a physical, biological or financial system is modeled by
a transformation T acting on a phase space X where time evolution
corresponds to the application of the map, moving an initial point in phase
space to a subsequent state. For example if x is the number of animals in
a population at time n=0 then a common population model predicts the next
population level at time n=1 to be T(x)=Kx(1-Lx), where
K,L are
parameters. The point T(x) then moves to T(T(x)):=T2(x). The sequence
{Tn(x)} corresponds to population levels at subsequent times
n.
More generally if φ: X→R is a function or measurement on the
system then {φ⋅Tn} is a time-series of measurements on the
underlying system. If the map T has an invariant probability measure then
this time-series {φ⋅Tn} is a stochastic process: that is, a
sequence of random variables on a probability space. Although the
time-series of measurements on such deterministic systems are not
independent (in fact usually highly correlated) the standard limit theorems
such as the strong law of large numbers, the central limit theorem and
Brownian motion-like behavior of the stochastic process often hold if the
system has a slight degree of hyperbolicity. We show how ideas and
techniques from probability and ergodic theory can be applied to understand
the statistical behavior of time-series of measurements on `chaotic'
dynamical systems.
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