We study entropies caused by the unstable part of partially hyperbolic systems. We define unstable metric entropy and unstable topological entropy, and establish a variational principle for partially hyperbolic diffeomorphsims, which states that the unstable topological entropy is the supremum of the unstable metric entropy taken over all invariant measures. The unstable metric entropy for an invariant measure is defined as a conditional entropy along unstable manifolds, and it turns out to be the same as that given by Ledrappier-Young, though we do not use increasing partitions. The unstable topological entropy is defined equivalently via separated sets, spanning sets and open covers along a piece of unstable leaf, and it coincides with the unstable volume growth along unstable foliation. We also obtain some properties for the unstable metric entropy such as affineness, upper semi-continuity and a version of Shannon-McMillan-Breiman theorem.

Let fS i gì D1 be an iterated function system (IFS) on R d with attractor K. Let . †; / denote the one-sided full shift over the alphabet f1; : : : ;`g. We define the projection entropy function h on the space of invariant measures on † associated with the coding map W † ! K and develop some basic ergodic properties about it. This concept turns out to be crucial in the study of dimensional properties of invariant measures on K. We show that for any conformal IFS (respectively, the direct product of finitely many conformal IFSs), without any separation condition, the projection of an ergodic measure under is always exactly dimensional and its Hausdorff dimension can be represented as the ratio of its projection entropy to its Lyapunov exponent (respectively, the linear combination of projection entropies associated with several coding maps). Furthermore, for any conformal IFS and certain affine IFSs, we prove a variational principle between the Hausdorff dimension of the attractors and that of projections of ergodic measures.

We consider a piecewise smooth expanding map f on the unit interval that has the form $f(x)=x+x^{1+\gamma}+o(x^{1+\gamma})$ near 0, where $0<\gamma < 1$. We prove by showing both lower and upper bounds that the rate of decay of correlations with respect to the absolutely continuous invariant probability measure $\mu$ is polynomial with the same degree $1/\gamma-1$ for Lipschitz functions. We also show that the density function h of $\mu$ has the order $x^{-\gamma}$ as $x\to 0$. Perron–Frobenius operators are the main tool used for proofs.

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Abstract. In this paper, using thermodynamic formalism for the sub-additive potential, upper bounds for the Hausdorff dimension and the box dimension of non-conformal repellers are obtained as the sub-additive Bowen equation. The map f only needs to be C 1 , without additional conditions. We also prove that all the upper bounds for the Hausdorff dimension obtained in earlier papers coincide. This unifies their results. Furthermore we define an average conformal repeller and prove that the dimension of an average conformal repeller equals the unique root of the sub-additive Bowen equation.

The purpose of this paper is to present some simple examples that are hyperbolic everywhere except at one point, but which do not admit SBR measures. Each example has a fixed point at which the larger eigenvalue is equal to one and the smaller eigenvalue is less than one.

Abstract. A diffeomorphism f of a compact manifold M is called "almost Anosov" if it is uniformly hyperbolic away from a finite set of points. We show that under some nondegeneracy condition, every almost Anosov diffeomorphism admits an invariant measure µ that has absolutely continuous conditional measures on unstable manifolds. The measure µ is either finite or infinite, and is called SBR measure or infinite SBR measure respectively. Therefore,

For a large class of nonuniformly expanding maps of R m , with indifferent fixed points and unbounded distorsion and non necessarily Markovian, we construct an absolutely continuous invariant measure. We extend to our case techniques previously used for expanding maps on quasi-Hölder spaces. We give general conditions and provide examples to which apply our result. IntroductionA challenge problem in smooth ergodic theory is to construct invariant measures for multidimensional maps T with some sort of weak hyperbolicity and then to study their statistical properties (decay of correlations, central limit theorem, distribution of return times, etc.). For nonuniformly expanding endomorphisms of R m , only few results exist at the moment. When the system has a Bernoulli structure, or verifies the so-called "finite range structure", and it enjoys a suitable distortion relation (Renyi's condition), M. Yuri [22,23] was able to construct an invariant, possibly σ-finite, measure absolutely continuous with respect to the Lebesgue measure. Young's tower ([20, 21]), which is mainly for nonuniformly hyperbolic systems, also works for nonuniformly expanding maps, and the invariant measures and other statistical properties can be obtained, if some bounded distortion properties are assumed. In Alves-Bonatti-Viana's work ([2], also see [3,4,5]), nonuniformly expansion are understood as the average value of log ||DT (x) −1 || along the orbits to be less than zero for almost every points. Under some conditions on the set of critical points, they can