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I have seen some attempt in considering topological pressure for Julia sets of exponential function, and elliptic function. However, there exists few reference according to my knowledge?

I want to ask whether there exist further general variational principle to make ergodic theory can been applied into transcendental dynamical system.

Any comments and reference will be appreciated?

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2 Answers

up vote 2 down vote accepted

There is quite a lot of literature these days on the measurable dynamics of transcendental entire functions. You may be interested, in particular, in the works of Mariusz Urbanski and his co-authors (in particular, his papers with Volker Mayer come to mind).

There is a very general result (http://arxiv.org/abs/1007.3855) regarding topological pressure and conformal measures by Baranski, Karpinska and Zdunik. This applies to all functions with a finite set of singular values, and to most functions in the Eremenko-Lyubich class $\mathcal{B}$. (I suspect that it extends to the whole class $\mathcal{B}$, in fact.)

Outside of the class $\mathcal{B}$, much less is known, and you have to be a little bit careful. An interesting case to consider may be Bishop's recent example of a function whose Julia set has Hausdorff dimension $1$. Here the hyperbolic dimension - and hence dimension of the 'radial Julia set' - is strictly less than $1$. The map is strongly expanding near its Julia set, so I would expect that some nice results would hold. I am not sure, however, that any of the known results would apply straightaway.

Finally, the situation is a little bit complicated, even within the class $\mathcal{B}$. Firstly, if you are going to study nice invariant measures, you will need to allow sigma-finite measures, even for function-theoretically very simple maps (see Dobbs and Skorulski, http://arxiv.org/abs/0801.0075).

On the other hand, for hyperbolic functions, it turns out to be often true that the hyperbolic dimension of the Julia set is strictly less than two, and there are nice conformal and invariant measures (see the papers by Urbanski and Mayer). However, I recently showed that this is not true in general, even for functions of finite order in the class $\mathcal{B}$; see my paper "Hyperbolic entire functions with full hyperbolic dimension ...", to appear in the Proceedings of the London Mathematical Society. This implies that this map cannot have a conformal measure on the radial Julia set (as this measure would otherwise have to agree with Lebesgue measure).

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Thank you for your elegant reply, professors. –  yaoxiao Oct 24 '13 at 17:57
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One possible reference is MR2304299 Gelfert, Katrin; Wolf, Christian Topological pressure for one-dimensional holomorphic dynamical systems. Bull. Pol. Acad. Sci. Math. 55 (2007), no. 1, 53–62 Some conditions are imposed on the behavior of the iterates of the holomorphic map outside a compact set.

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