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Let us consider a diffeomorphism of a compact real manifold (complex manifold defined over the reals), and let us say that the diffeomorphism is birational. Hence, it extends to a birational map from the complex manifold, and we can associate to it the complex entropy /real entropy as well as Julia/Fatou sets.

I presume that there is a relation between the real topological entropy and the real points of the Fatou/Julia sets. Can someone give me a reference where I can find such a relation?

For instance, is there an equivalence which says that the real topological entropy is trivial if and only if there are many real points in the Fatou set is large ? (with of course a precise notion of "many", for example with density?)

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  • $\begingroup$ What is a Fatou/Julia set of a diffeomorphism of a real manifold?? Could you define, or give a reference? $\endgroup$ Commented Aug 23, 2013 at 5:44
  • $\begingroup$ To be simple, I edited the question to have something where the Fatou/Julia set are classically defined. $\endgroup$ Commented Aug 23, 2013 at 8:33

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I do not know what is a Fatou/Julia set of a diffeomorphism of a real manifold.

The usual definitions of Fatou/Julia sets are applied to rational map of a 1-dimensional complex manifold, namely the Riemann sphere.

The topological entropy of such map is always equal to log of the degree, and this is not related to its Fatou or Julia set.

EDIT Aug 23: Similar situation persists in higher dimension (in holomorphic setting) MR2026895 Gromov, Mikhail, On the entropy of holomorphic maps. Enseign. Math. (2) 49 (2003), no. 3-4, 217–235. The topological entropy of a surjective holomorphic self-map of the complex projective space is equal to the logarithm of the topological degree.

Again, this has nothing to do with Fatou/Julia sets.

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  • $\begingroup$ In the last sentence, you meant "topological entropy" instead of "topological degree", right? And there are generalizations of Fatou/Julia theory to real manifolds. Whether one finds them interesting or otherwise is another matter, but they are out there. $\endgroup$ Commented Aug 23, 2013 at 15:04
  • $\begingroup$ Margaret, thanks for the correction. There ARE attempts to extend the Fatou-Julia theory to some real manifolds, like $R^n$ or $S^n$ but there is no generally accepted definition of the Julia set in such generality yet. At least I don't know of such a definition. So I think the author of the question should state preceisely what he means. $\endgroup$ Commented Aug 23, 2013 at 20:24
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An attempt to extend the Fatou-Julia theory to continuous self-maps of a connected Riemannian manifold $M$ with a $\mathcal{C}^\infty$ smooth metric was made in the following book: MR1784605 (2001i:37001) Hu, Pei-Chu; Yang, Chung-Chun: Differentiable and complex dynamics of several variables. (English summary) Mathematics and its Applications, 483. Kluwer Academic Publishers, Dordrecht, 1999. x+338 pp. ISBN: 0-7923-5771-X

Like the classical theory in $\mathbb{CP}^1$ or $\mathbb{C}$, it uses the notion of a normal family, somewhat modified for the purpose. A family $\mathcal{F}$ of such maps is called normal on $M$ if every sequence in $\mathcal{F}$ contains a subsequence which either is relatively compact in $\mathcal{C}(M,M)$ or compactly divergent. Theorem 1.10 says that for an $f \in \mathcal{C}(M,M)$ there is a maximal open subset $F(f)$ of $M$ on which the family $\mathcal{F}=\{f^n: n=1,2,...\}$ is normal. This set is called the Fatou set and its complement is the Julia set. For interesting phenomena to happen, one usually needs additional assumptions, but these are not too restrictive. The relation between the topological entropy of the map and Julia sets is discussed in \S 1.7, while \S 1.6 discusses the relation between topological degree and Julia sets. Unfortunately, I do not remember precise statements and do not have the book at hand (the Google preview stops right there), but I hope this will get you started in the right direction.

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  • $\begingroup$ I realized I answered the old question. The new one is radically different. For the study of entropy of birational maps, see e.g. MR2677899 Bedford, Eric; Kim, Kyounghee Continuous families of rational surface automorphisms with positive entropy. Math. Ann. 348 (2010), no. 3, 667–688, or other joint publications of these authors (but you probably know these already, judging from your research interests). $\endgroup$ Commented Aug 23, 2013 at 16:56
  • $\begingroup$ Thanks for the information. Your definition of Fatou-Julia set is the one I had in mind. It seems to work in general. I would be happy to have a relation between dense Fatou set (or large one) and null entropy in any reasonable case. And of course I know the article you mention, but I dont think it answer this. $\endgroup$ Commented Aug 25, 2013 at 19:48
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Roughly speaking, there is close relation between topology entropy and expanding point in the dynamics system. Entropy is mainly depend on whether the dynamical behaviour on Julia sets is complicate or not.

For example, for a (weakly) contracting system $(X,T,d)$, $$d(Tx, Ty)\leq d(x,y)$$, the topological entropy is $0$. (T=id is an special example)

An another example, $A$ is a linear map from $R^{n}$ to $R^{n}$, it entropy is equal to $$\sum\log |\lambda_{i}|^{+}$$. We can see there is no contribution for the contracting eigen-direction in computing topological entropy.

In many cases, we know Julia sets is the closure of repelling periodic points. for rational function dynamics system, by a well known theorem of Sullivan, we know there exist no wandering Fatou domain. It implies all Fatou component are periodic and preperiodic. For (pre)periodic Fatou component, we have a complete classification for it. attracting basin, parabolic basin, Hermann ring, Siegel disk. dynamical behavior on fatou sets is not chaotic. However entropy gives information of chaotic behavior of a system. In some sense, there are little contributions for Fatou sets in computing topological entropy.

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  • $\begingroup$ Thanks for the informations. However, I was hoping for something more explicit, like a criterion given by the Fatou set to decide wether or not the map has non-trivial entropy. $\endgroup$ Commented Aug 22, 2013 at 17:19

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