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Mathematical Physics Seminar

Roger Nussbaum - Hidden Positivity and High Order Rigorous Approximation of Hausdorff Dimension

Roger Nussbaum

Location:  Hill Center room 705
Date & time: Thursday, 27 February 2020 at 2:00PM - 3:00PM

  

Roger Nussbaum - Rutgers University

  Date/Time/Location
Thursday, February 27th, 2:00pm; Hill 705

Title
" Hidden Positivity and High Order Rigorous Approximation of Hausdorff Dimension "

Given a complete metric space X with metric s and a finite set of "conformal" contraction mappings f_i :X--->X, i=1,2,...n, there is a unique, nonempty, invariant compact set C such that C equals the union of the sets f_i(C). It is of interest to estimate the Hausdorff dimension of C. A much-studied example from number theory is provided by X=[0, infinity) and f_i (x) =1/(x+m_i), where m_1, m_2, ...,m_n are distinct positive integers. We shall describe how ideas from the theory of linear, positive operators (where positivity is understood in the sense of cone-preserving), a priori bounds on the derivatives of positive eigenvectors and tools from numerical analysis provide new insights and high order estimates of Hausdorff dimension. This is joint work with Professor Richard Falk.

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