Villanueva Segovia, Cristina (2018). Properties of Lipschitz quotient mappings on the plane. University of Birmingham. Ph.D.

Villanueva_Segovia18PhD.pdf
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Abstract
In the present work, we are concerned with the relation between the Lipschitz and coLipschitz constants of a mapping f : ℝ2 → ℝ2 and the cardinality of the inverse image of a point under the mapping f, depending on the norm on ℝ2.
It is known that there is a scale of real numbers 0 < ... < Pn <…< P1 < 1 such that for any Lipschitz quotient mapping from the Euclidean plane to itself, if the ratio between the coLipschitz and Lipschitz constants of f is bigger than Pn, then the cardinality of any fibre of f is less than or equal to n. Furthermore, it is proven that for the Euclidean case the values of this scale are Pn = 1/n + 1) for each n ∈ ℕ and that these are sharp.
A natural question is: given a normed space (ℝ2 , II · II) whether it is possible to find the values of the scale 0 < . . . < pn II · II < ... < p1 II · II < 1 such that for any Lipschitz quotient mapping from (ℝ2, II · II) to itself, with Lipschitz and coLipschitz constants equal to L and c respectively, the relation c/L > pn II · II implies #f 1 (x) ≤ n for all x ∈ ℝ2. In this work we prove that the same "Euclidean scale", Pn = 1/(n+1), works for any norm on the plane. Here we follow the general idea in a previous paper by Maleva but verify details carefully. On the other hand, the question whether this scale is sharp leads to different conclusions. We show that for some nonEuclidean norms the "Euclidean scale" is not sharp, but there are also nonEuclidean norms for which a Lipschitz quotient exists satisfying max# f  1(x) = 2 and c/L = 1/2.
Type of Work:  Thesis (Doctorates > Ph.D.)  

Award Type:  Doctorates > Ph.D.  
Supervisor(s): 


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College/Faculty:  Colleges (2008 onwards) > College of Engineering & Physical Sciences  
School or Department:  School of Mathematics  
Funders:  Other  
Other Funders:  Mexican Secretariat of Public Education  
Subjects:  Q Science > QA Mathematics  
URI:  http://etheses.bham.ac.uk/id/eprint/8266 
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