1,100 research outputs found
A variational approach to modeling slow processes in stochastic dynamical systems
The slow processes of metastable stochastic dynamical systems are difficult
to access by direct numerical simulation due the sampling problem. Here, we
suggest an approach for modeling the slow parts of Markov processes by
approximating the dominant eigenfunctions and eigenvalues of the propagator. To
this end, a variational principle is derived that is based on the maximization
of a Rayleigh coefficient. It is shown that this Rayleigh coefficient can be
estimated from statistical observables that can be obtained from short
distributed simulations starting from different parts of state space. The
approach forms a basis for the development of adaptive and efficient
computational algorithms for simulating and analyzing metastable Markov
processes while avoiding the sampling problem. Since any stochastic process
with finite memory can be transformed into a Markov process, the approach is
applicable to a wide range of processes relevant for modeling complex
real-world phenomena
Density of periodic sources in the boundary of a basin of attraction for iteration of holomorphic maps, geometric coding trees technique
We prove that if A is the basin of immediate attraction to a periodic
attracting or parabolic point for a rational map f on the Riemann sphere, then
periodic points in the boundary of A are dense in this boundary. To prove this
in the non simply- connected or parabolic situations we prove a more abstract,
geometric coding trees version
Statistical properties of topological Collet-Eckmann maps
We study geometric and statistical properties of complex rational maps
satisfying the Topological Collet-Eckmann Condition. We show that every such a
rational map possesses a unique conformal probability measure of minimal
exponent, and that this measure is non-atomic, ergodic and that its Hausdorff
dimension is equal to the Hausdorff dimension of the Julia set. Furthermore, we
show that there is a unique invariant probability measure that is absolutely
continuous with respect to this conformal measure, and we show that this
measure is exponentially mixing (it has exponential decay of correlations) and
that it satisfies the Central Limit Theorem.
We also show that for a complex rational map f the existence of such an
invariant measure characterizes the Topological Collet-Eckmann Condition, and
that this measure is the unique equilibrium state with potential - HD(J(f)) ln
|f'|
- …