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**Polynomial diffeomorphisms of \(\mathbb{C}^ 2\). IV: The measure of maximal entropy and laminar currents.**
*(English)*
Zbl 0792.58034

For the study of invertible, holomorphic dynamical systems, the simplest examples with interesting behavior are the polynomial diffeomorphisms of \(\mathbb{C}^ 2\). The paper under review is the fourth in a series initiated by two of the present authors [see E. Bedford and J. Smillie, ibid. 103, No. 1, 69-99 (1991; Zbl 0721.58037), J. Am. Math. Soc. 4, No. 4, 657-679 (1991; Zbl 0744.58068), Math. Ann. 294, No. 3, 395-420 (1992; Zbl 0765.58013)].

Corresponding to a Julia set in one dimension, for diffeomorphisms of \(\mathbb{C}^ 2\) three analogs are defined. The techniques of Fatou and Julia used in one dimension are based on Montel’s theory of normal families and do not readily generalize to higher dimensions. Later, it was seen that potential theory is an alternative tool to prove basic facts of Fatou- Julia theory. Potential theory has a natural extension to several complex variables.

The authors combine the methods of potential theory with tools from ergodic theory, especially Pesin’s theory of non-uniform hyperbolicity. The first crucial result is the following: The harmonic measure \(\mu\) is the unique measure of maximal entropy for a polynomial diffeomorphism of \(\mathbb{C}^ 2\).

It is shown that are sets, called Pesin boxes, on which \(\mu\) has a local product structure and their union has full \(\mu\) measure. For doing this, the holonomy map along the stable/unstable manifolds is studied; the conditional measures of \(\mu\) are preserved by the holonomy map.

Next, the primary tools are the theory of currents and the Ahlfors covering theorem. It is proved that \(\mu^ \pm\) are laminar currents, where the harmonic measure \(\mu\) and the currents \(\mu^ \pm\) are related by \(\mu = \mu^ + \wedge \mu^ -\). The holonomy map preserves the slices of \(\mu^ +\). Applications to the study of saddle points and to real Hénon mappings are given.

An appendix outlines an alternative sequence in which the results of this paper can be obtained. This alternate approach starts with Pesin theory and then proceeds to the theory of currents. The main difference is that the use of the methods of entropy theory is delayed until the end.

Corresponding to a Julia set in one dimension, for diffeomorphisms of \(\mathbb{C}^ 2\) three analogs are defined. The techniques of Fatou and Julia used in one dimension are based on Montel’s theory of normal families and do not readily generalize to higher dimensions. Later, it was seen that potential theory is an alternative tool to prove basic facts of Fatou- Julia theory. Potential theory has a natural extension to several complex variables.

The authors combine the methods of potential theory with tools from ergodic theory, especially Pesin’s theory of non-uniform hyperbolicity. The first crucial result is the following: The harmonic measure \(\mu\) is the unique measure of maximal entropy for a polynomial diffeomorphism of \(\mathbb{C}^ 2\).

It is shown that are sets, called Pesin boxes, on which \(\mu\) has a local product structure and their union has full \(\mu\) measure. For doing this, the holonomy map along the stable/unstable manifolds is studied; the conditional measures of \(\mu\) are preserved by the holonomy map.

Next, the primary tools are the theory of currents and the Ahlfors covering theorem. It is proved that \(\mu^ \pm\) are laminar currents, where the harmonic measure \(\mu\) and the currents \(\mu^ \pm\) are related by \(\mu = \mu^ + \wedge \mu^ -\). The holonomy map preserves the slices of \(\mu^ +\). Applications to the study of saddle points and to real Hénon mappings are given.

An appendix outlines an alternative sequence in which the results of this paper can be obtained. This alternate approach starts with Pesin theory and then proceeds to the theory of currents. The main difference is that the use of the methods of entropy theory is delayed until the end.

Reviewer: I.Mihai (Bucureşti)

### MSC:

37F10 | Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets |

32H50 | Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables |

58C35 | Integration on manifolds; measures on manifolds |

58A25 | Currents in global analysis |

### Keywords:

polynomial diffeomorphisms; potential theory; Fatou-Julia theory; ergodic theory; harmonic measure; laminar currents; Pesin theory
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\textit{E. Bedford} et al., Invent. Math. 112, No. 1, 77--125 (1993; Zbl 0792.58034)

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