engleski

Multiplicity of Fixed Points and Growth of r- Neighborhoods of Orbits

In this joint work with Pavao Marde\v si\' c, University of Bourgogne, and Vesna \v Zupanovi\' c, University of Zagreb, we study the relationship between the multiplicity of a fixed point of a function $g$, and the dependence on $\varepsilon$ of the length of $\varepsilon$-neighborhood of any orbit of $g$, tending to the fixed point. In case $g$ is differentiable at the fixed point, the relationship was discovered before by Elezovi\' c, \v Zubrini\' c and \v Zupanovi\' c. Moreover, Minkowski contents and box dimension of the orbit were used as appropriate tools for describing the growth of $\varepsilon-$neighborhood. We generalize this result to a special class of non- differentiable functions which have a development in a Chebyshev scale. We recover the relationship between multiplicity of fixed points and the dependence on $\varepsilon$ of the length of $\varepsilon$-neighborhoods of orbits. In this case, box dimension is not a precise measure for describing growth of $\varepsilon-$neighbourhoods, so the notion of generalized Minkowski contents and generalized Minkowski order with respect to the Cebyshev scale is introduced instead. The notion is greatly motivated by the idea of generalized Minkowski contents and gauge functions introduced earlier by Lapidus and He. (lectured by Maja Resman)

limit cycles; multiplicity; cyclicity; Chebyshev scale; critical Minkowski order; box dimension; homoclinic loop

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