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Complex maps without invariant densities

Henk Bruin·Mike Todd
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Abstract

We consider complex polynomials $f(z) = z^\ell+c_1$ for $\ell \in 2\N$ and $c_1 \in \R$, and find some combinatorial types and values of $\ell$ such that there is no invariant probability measure equivalent to conformal measure on the Julia set. This holds for particular Fibonacci-like and Feigenbaum combinatorial types when $\ell$ sufficiently large and also for a class of `long-branched' maps of any critical order.

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Henk Bruin, Mike Todd. 2006-11-27. Complex maps without invariant densities. https://doi.org/10.1088/0951-7715%2F19%2F12%2F012

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