Complex maps without invariant densities

Details Complex maps without invariant densities Complex maps without invariant densities

2006-06-27

arxiv.org/abs/math/0606677v2

Nonlinearity 19 (2006) 2929-2945

Mathematics

Dynamical Systems

Typos corrected, minor changes, principally to Section 6

Scientific paper

10.1088/0951-7715/19/12/012

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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