Unique Measure Of Maximal Entropy Research Articles

Distribution of periodic points of polynomial diffeomorphisms of C2

This paper deals with the dynamics of a simple family of holomorphic diffeomorphisms of $\C^2$: the polynomial automorphisms. This family of maps has been studied by a number of authors. We refer to [BLS] for a general introduction to this class of dynamical systems. An interesting object from the point of view of potential theory is the equilibrium measure $\mu$ of the set $K$ of points with bounded orbits. In [BLS] $\mu$ is also characterized dynamically as the unique measure of maximal entropy. Thus $\mu$ is also an equilibrium measure from the point of view of the thermodynamical formalism. In the present paper we give another dynamical interpretation of $\mu$ as the limit distribution of the periodic points of $f$.

Polynomial diffeomorphisms ofC 2. IV: The measure of maximal entropy and laminar currents

This paper concerns the dynamics of polynomial automorphisms of ${\bf C}^2$. One can associate to such an automorphism two currents $\mu^\pm$ and the equilibrium measure $\mu=\mu^+\wedge\mu^-$. In this paper we study some geometric and dynamical properties of these objects. First, we characterize $\mu$ as the unique measure of maximal entropy. Then we show that the measure $\mu$ has a local product structure and that the currents $\mu^\pm$ have a laminar structure. This allows us to deduce information about periodic points and heteroclinic intersections. For example, we prove that the support of $\mu$ coincides with the closure of the set of saddle points. The methods used combine the pluripotential theory with the theory of non-uniformly hyperbolic dynamical systems.

Mahler measure and entropy for commuting automorphisms of compact groups

We compute the joint entropy ofd commuting automorphisms of a compact metrizable group. LetR d = ℤ[u 1 ±1 ,...,[d 1 ±1 ] be the ring of Laurent polynomials ind commuting variables, andM be anR d -module. Then the dual groupX M ofM is compact, and multiplication onM by each of thed variables corresponds to an action α M of ℤ d by automorphisms ofX M . Every action of ℤ d by automorphisms of a compact abelian group arises this way. Iff∋R d , our main formula shows that the topological entropy of $$\alpha _{R_d /\left\langle f \right\rangle } $$ is given by $$h\left( {\alpha _{R_d /\left\langle f \right\rangle } } \right) = \log M\left( f \right) = \int\limits_0^1 \ldots \int\limits_0^1 {\log \left| {f\left( {e^{2\pi it_1 } , \ldots , e^{2\pi it_d } } \right)} \right|} dt_1 \ldots dt_d $$ , where M(f) is the Mahler measure off. This reduces to the classical result for toral automorphisms via Jensen's formula. While the entropy of a single automorphism of a compact group is always the logarithm of an algebraic integer, this no longer seems to hold for joint entropy of commuting automorphisms since values such as 7ζ(3)/4π2 occur. If p is a non-principal prime ideal, we show $$h\left( {\alpha _{R_d /p} } \right) = 0$$ . Using an analogue of the Yuzvinskii-Thomas addition formula, we computeh(α M ) for arbitraryR d -modulesM, and then the joint entropy for an action of ℤ d on a (not necessarily abelian) compact group. Using a result of Boyd, we characterize those α M which have completely positive entropy in terms of the prime ideals associated toM, and show this condition implies that α M is mixing of all orders. We also establish an analogue of Berg's theorem, proving that if α M has finite entropy then Haar measure is the unique measure of maximal entropy if and only if α M has completely positive entropy. Finally, we show that for expansive actions the growth rate of the number of periodic points equals the topological entropy.

Entropy properties of rational endomorphisms of the Riemann sphere

AbstractIn this paper the existence of a unique measure of maximal entropy for rational endomorphisms of the Riemann sphere is established. The equidistribution of pre-images and periodic points with respect to this measure is proved.

On the distribution of periodic points for β-shifts

It is shown that for β-shifts the periodic points are uniformly distributed with respect to the unique measure of maximal entropy, and that the invariant measures with support on a single periodic orbit are dense in the space of all invariant measures.