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Abstract:
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This thesis presents a method for calculating the polynomial entropy of the topolog-
ical dynamic system with finitely many non-wandering points. A special coding is adapted for
such systems. Thanks to this coding the polynomial entropy can be bounded by the number
of specific mutually singular points in the closures of stable manifolds of non-wandering points.
This method was applied to Morse gradient systems. It is shown that the polynomial entropy
of the Morse gradient system is bounded by n(F ) − 1, where n(F ) is the number of different
Morse indices of critical points of the Morse function F. If Morse gradient systems on mani-
folds of dimension n has only critical points of indices 0 and n, it is proved that the polynomial
entropy is equal to 1, and if the system has critical points of indices 0, n/2 and n, it is proved
that polynomial entropy is equal to 2. The polynomial entropy for different parameter values
in logistic map has also been calculated, and it has been shown that the polynomial entropy
distinguishes the systems of low complexity with drastically different behaviours, which cannot
be distinguished by the topological entropy. The example of the homeomorphism of the con-
nected compact metric space that is not equicontinuous and with vanishing polynomial entropy
is also given. |