Pointwise Variation Growth and Entropy of the Descartes Product of a Few of Interval Maps

Risong Li; Zengxiong Cheng
October 2011
Pure Mathematics;Oct2011, Vol. 1 Issue 3, p184
Academic Journal
In this paper, the definition of pointwise variation growth of interval maps was extended to continuous self-maps on k-dimensional space I1 x I2 x�x Ik, where Ii is a closed interval. Let fi : Ii & larr; Ii be a continuous map and the total variation Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. be bounded for all n = 0, i = 1,2,�k. It was proved that the inequality ? ((x1,x2,�,xk), f1xf2,f2x�xfk) > s ((x1,x2,�,xk), f1xf2,f2x�xfk holds for any (x1,x2,�,xk), ? I1 x I2 x�xIk and that the functions Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. and Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed., which maps a point (x1,x2,�,xk) to its local growth rate of variation and its local topological entropy respectively, are both upper semi-continnuous. The variational princeple which is corresponding to a variational principle on mappings on an interval was obtained. When the map fi ? Ii is topologically transitive, i = 1, 2, �, k, the corresponding ii i result of mappings on an interval was also extended.


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