The simulation of random processes on digital computers: Unavoidable order

The Simulation of Random Processes on Digital Computers with Čebyšev Mixing Transformations

Shadowing and iterative interpolation for Čebyšev mixing transformations

In this paper, we investigate metric properties and dispersive effects of strongly mixing transformations on general metric spaces endowed with a finite measure; in particular, we investigate their connections with the theory of generalized (geometric) diameters on general metric spaces. We first show that the known result by Rice [17,Theorem 2] (motivated by some physical phenomena and offer some clarifications of these phenomena), which is a substantial improvement of Theorems 1 and 2 due to Erber, Schweizer and Sklar [4], can be generalized in such a way that this result remains valid when “ordinary diameter” is replaced by “geometric diameter of any finite order”. Next we show that “ordinary essential diameter” in the mentioned Rice’s result can be replaced by “essential geometric diameter of any finite order”. These results also complement the previous results of Fatkic [ 6,8,10], Saff [18] and Sempi [20].

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Strongly mixing transformations and geometric diameters

We complete the picture how the asymptotic behavior of a dynamical system is reflected by properties of the associated Perron-Frobenius operator. Our main result states that strong convergence of the powers of the Perron-Frobenius operator is equivalent to setwise convergence of the underlying dynamic in the measure algebra. This situation is furthermore characterized by a uniform mixing-like property of the system.

R. E. Rice

Papers

On mixing transformations