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Full-Text Articles in Physical Sciences and Mathematics
Geometric And Measure-Theoretic Shrinking Targets In Dynamical Systems, Joseph Rosenblatt, Mrinal Kanti Roychowdhury
Geometric And Measure-Theoretic Shrinking Targets In Dynamical Systems, Joseph Rosenblatt, Mrinal Kanti Roychowdhury
School of Mathematical and Statistical Sciences Faculty Publications and Presentations
We consider both geometric and measure-theoretic shrinking targets for ergodic maps, investigating when they are visible or invisible. Some Baire category theorems are proved, and particular constructions are given when the underlying map is fixed. Open questions about shrinking targets are also described.
Optimal Quantization Via Dynamics, Joseph Rosenblatt, Mrinal Kanti Roychowdhury
Optimal Quantization Via Dynamics, Joseph Rosenblatt, Mrinal Kanti Roychowdhury
School of Mathematical and Statistical Sciences Faculty Publications and Presentations
Quantization for probability distributions refers broadly to estimating a given probability measure by a discrete probability measure supported by a finite number of points. We consider general geometric approaches to quantization using stationary processes arising in dynamical systems, followed by a discussion of the special cases of stationary processes: random processes and Diophantine processes. We are interested in how close stationary process can be to giving optimal n-means and nth optimal mean distortion errors. We also consider different ways of measuring the degree of approximation by quantization, and their advantages and disadvantages in these different contexts.