Assume that T is a conservative ergodic measure-preserving transformation of the infinite measure space $(X,\mathcal{A},\mu)$. We study the asymptotic behaviour of occupation times of certain subsets of infinite measure. Specifically, we prove a Darling–Kac type distributional limit theorem for occupation times of barely infinite components which are separated from the rest of the space by a set of finite measure with continued-fraction (CF)-mixing return process. In the same setup we show that the ratios of occupation times of two components separated in this way diverge almost everywhere. These abstract results are illustrated by applications to interval maps with indifferent fixed points.

We extend the Delorme–Guichardet characterization of Kazhdan property T groups to r-discrete measured groupoids. We give several applications, in particular to the stability results of Kazhdan property T groupoids and to the study of cocycles taking values in a group having the Haagerup property.

We study the discrete quasi-periodic Schrödinger equation \[-(u_{n+1}+u_{n-1})+\lambda V(\theta+n\omega)u_n=Eu_n\] with a non-constant C1 potential function $V:\mathbb{T}\to\mathbb{R}$. We prove that for sufficiently large $\lambda$ there is a set $\Omega\subset\mathbb{T}$ of frequencies $\omega$, whose measure tends to 1 as $\lambda\to\infty$, with the following property. For each $\omega\in\Omega$ there is a ‘large’ (in measure) set of energies E, all lying in the spectrum of the associated Schrödinger operator (and hence giving a lower estimate on the measure of the spectrum), such that the Lyapunov exponent is positive and, moreover, the projective dynamical system induced by the Schrödinger cocycle is minimal but not ergodic.

In this paper we consider the family of rational maps of the complex plane given by \[z^2+\frac{\lambda}{z^2}\] where $\lambda$ is a complex parameter. We regard this family as a singular perturbation of the simple function $z^2$. We show that, in any neighborhood of the origin in the parameter plane, there are infinitely many open sets of parameters for which the Julia sets of the corresponding maps are Sierpinski curves. Hence all of these Julia sets are homeomorphic. However, we also show that parameters corresponding to different open sets have dynamics that are not conjugate.

We prove structure theorems for holomorphic correspondences realizing matings between pinched polynomial-like maps with connected Julia sets and Hecke groups (Fuchsian representations of free products C2 * Cn of cyclic groups of orders 2 and n). We show that such matings are generated by two-to-two subcorrespondences if and only if the maps are Chebyshev-like. We describe the dynamical behaviour of these matings in detail in the case n = 4 and we present a conjectured combinatorial description of the connectedness locus in parameter space in this case.

We define and study new invariants called pre-image entropies which are similar to the standard notions of topological and measure-theoretic entropies. These new invariants are only non-zero for non-invertible maps, and they give a quantitative measurement of how far a given map is from being invertible. We obtain analogs of many known results for topological and measure-theoretic entropies. In particular, we obtain product rules, power rules, analogs of the Shannon–Breiman–McMillan theorem, and a variational principle.

We consider zero-noise limits of random perturbations of dynamical systems and examine, in terms of the continuity of entropy and Lyapunov exponents, circumstances under which these limits are Sinai–Ruelle–Bowen (SRB) measures. The ideas of this general discussion are then applied to specific classes of attractors. We prove, for example, that partially hyperbolic attractors with one-dimensional center subbundles always admit SRB measures.

We study the family of quadratic maps fa(x) = 1 − ax2 on the interval [−1, 1] with 0 [nle ] a [nle ] 2. When small holes are introduced into the system, we prove the existence of an absolutely continuous conditionally invariant measure using the method of Markov extensions. The measure has a density which is bounded away from zero and is analogous to the density for the corresponding closed system. These results establish the exponential escape rate of Lebesgue measure from the system, despite the contraction in a neighborhood of the critical point of the map. We also prove convergence of the conditionally invariant measure to the SRB measure for fa as the size of the hole goes to zero.

Let X be a compact metric space, f a continuous transformation on X, and Y a vector space with linear compatible metric. Denote by M(X) the collection of all the probability measures on X. For a positive integer n, define the nth empirical measure $L_{n}:X\mapsto M(X)$ as

\[ L_{n}x=\frac{1}{n}\sum _{k=0}^{n-1}\delta_{f^{k}x}, \]

where $\delta_{x}$ denotes the Dirac measure at x. Suppose $\Xi:M(X)\mapsto Y$ is continuous and affine with respect to the weak topology on M(X). We think of the composite

\[ \Xi\circ L_{n}:X \xrightarrow{L_n} M(X)\xrightarrow{\Xi} Y \]

as a continuous and affine deformation of the empirical measure Ln. The set of divergence points of such a deformation is defined as

\[ D(f,\Xi)=\{x\in X\mid \mbox{the limit of }\Xi L_n x \mbox{ does not exist}\}. \]

In this paper we show that for a continuous transformation satisfying the specification property, if $\Xi(M(X))$ is a singleton, then set of divergence points is empty, i.e. <formula form="inline" disc="math" id="ffm008"><formtex notation="AMSTeX">$D(f,\Xi)=\emptyset$, and if $\Xi(M(X))$ is not a singleton, then the set of divergence points has full topological entropy, i.e.

\[ h_{\rm top}(D(f,\Xi))=h_{\rm top}(f). \]

We find an upper bound for the entropy of a systolically extremal surface, in terms of its systole. We combine the upper bound with Katok's lower bound in terms of the volume, to obtain a simpler alternative proof of Gromov's asymptotic estimate for the optimal systolic ratio of surfaces of large genus. Furthermore, we improve the multiplicative constant in Gromov's theorem. We show that every surface of genus at least 20 is Loewner. Finally, we relate, in higher dimension, the isoembolic ratio to the minimal entropy.

In this paper, we establish analytic conjugacy theorems of Poincaré type and Poincaré–Dulac type for analytic random dynamical systems based on their Lyapunov exponents.

Hill's lunar problem appears in celestial mechanics as a limit of the restricted three-body problem. It is parameter-free and thus globally far from any simple well-known problem, and has shed strong numerical evidence of its lack of integrability in the past. An algebraic proof of meromorphic non-integrability is presented here. Beyond the result itself, the paper can also be considered as an example of the application of differential Galois and Morales–Ramis theories to a significant problem.

We prove pointwise ergodic theorems for general families of radial averages on semi-simple Lie groups, as well as strong Lp-maximal inequalities for the corresponding maximal functions, for p > 1. The methods of proof make essential use of exponential volume growth on the group, and of spectral considerations utilizing the existence of spectral gaps in measure-preserving actions of simple groups of real rank at least two. We also consider the case of the action by convolutions on the symmetric space, where we establish maximal inequalities of weak-type, going beyond Lp, p > 1. The proof here is based on some new convolution inequalities which are established by direct geometric comparison arguments. These inequalities can be viewed as a quantitative version of the wave front lemma. The maximal inequalities and pointwise ergodic theorems we establish are in sharp contrast to what one would be lead to expect by comparison with the analogous averages on the Abelian groups $\mathbb{R}^n$. We explain how the contrast arises from the difference between the polynomial volume growth on $\mathbb{R}^n$ and the exponential volume growth on semi-simple groups.

We study the dynamics of compositions of a sequence of holomorphic mappings in $\mathbb{P}^k$. We define ergodicity and mixing for non-autonomous dynamical systems, and we construct totally invariant measures for which our sequence satisfies these properties.

In this paper we use infinite ergodic theory to study limit sets of essentially free Kleinian groups which may have parabolic elements of arbitrary rank. By adapting a method of Adler, we construct a section map S for the geodesic flow on the associated hyperbolic manifold. We then show that this map has the Markov property and that it is conservative and ergodic with respect to the invariant measure induced by the Liouville–Patterson measure. Furthermore, we obtain that S is rationally ergodic with respect to different types of return sequences (an), which are governed by the exponent of convergence $\delta$ and the maximal possible rank kmax of the parabolic elements of the group as follows \[a_n\asymp\begin{cases}n^{2\delta-k_{\max}}&\text{for }\delta<(k_{\max}+1)/2,\\n/\!\log n&\text{for }\delta=(k_{\max}+1)/2,\\n&\text{for }\delta>(k_{\max}+1)/2.\end{cases}\] Subsequently, we give a discussion of an associated canonical map T, which is an analogue of the Bowen–Series map in the Fuchsian case. We show that T is pointwise dual ergodic with respect to these return sequences (an), which then allows us to determine the index of variation $\beta=\min\{1,2\delta-k_{\max}\}$, and to deduce that the ergodic sums Sn(f)/an converge strongly distributional to the Mittag–Leffler distribution of index $\beta$. We then give applications to number theory and to the statistics of cuspidal windings. Also, as a corollary we obtain a special case of Sullivan's result that the geodesic flow on a geometrically finite hyperbolic manifold is ergodic with respect to the Liouville–Patterson measure.

In this paper, we consider piecewise smooth dynamical systems with generalized indifferent periodic orbits associated to a given potential function. We show that the presence of such orbits causes non-uniqueness of equilibrium states (phase transitions) and non-Gibbsianness of equilibrium measures. If, in addition, the system exhibits local exponential instability, we show that the distribution of the stopping times over hyperbolic regions has powerlike tails with respect to any s-conformal measure, where s is the smallest zero of Bowen's equation. Finally, we prove that the Hausdorff dimension of the limit set is equal to s provided the dynamics is piecewise conformal.

For countable-to-one transitive Markov maps, we show that the natural extensions of invariant ergodic weak Gibbs measures absolutely continuous with respect to weak Gibbs conformal measures possess a version of the u-Gibbs property. In particular, if dynamical potentials admit generalized indifferent periodic points then the natural extensions exhibit a non-Gibbsian character in statistical mechanics. Our results can be applicable to certain non-hyperbolic number-theoretical transformations of which natural extensions possess unstable (respectively stable) leaves with subexponential expansion (respectively contraction).

We prove a non-conventional pointwise convergence theorem for a nilsystem, and give an explicit formula for the limit.