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arXiv · 1906.03882

Distribution of typical orbits for a skew-product map generated by random dynamics of finitely many rational maps

Shrihari Sridharan·Sharvari Neetin Tikekar·Atma Ram Tiwari
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Abstract

In this paper, we consider the dynamics of a skew-product map defined on the Cartesian product of the symbolic one-sided shift space on $N$ symbols and the complex sphere where we allow $N$ rational maps, $R_{1}, R_{2}, \cdots, R_{N}$, each with degree $d_{i};\ 1 \le i \le N$ and with at least one $R_{i}$ in the collection whose degree is at least $2$. We obtain results regarding the distribution of pre-images of points and the periodic points in a subset of the product space (where the skew-product map does not behave normally). We further explore the ergodicity of the Sumi-Urbanskii (equilibrium) measure associated to some real-valued H\"{o}lder continuous function defined on the Julia set of the skew-product map and obtain estimates on the mean deviation of the behaviour of typical orbits, violating such ergodic necessities.

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Shrihari Sridharan, Sharvari Neetin Tikekar, Atma Ram Tiwari. 2019-06-10. Distribution of typical orbits for a skew-product map generated by random dynamics of finitely many rational maps. https://arxiv.org/abs/1906.03882

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