We show that the Morse system is approximately transitive (AT) and that a system of positive entropy cannot be AT. We give examples of zero-entropy systems that are not AT, one of which is a two-point extension of a system that is AT.

We prove the L2-convergence for the linear multiple ergodic averages of commuting transformations $T_{1}, \dots, T_{l}$, assuming that each map Ti and each pair TiTj-1 is ergodic for $i\neq j$. The limiting behavior of such averages is controlled by a particular factor, which is an inverse limit of nilsystems. As a corollary we show that the limiting behavior of linear multiple ergodic averages is the same for commuting transformations.

A one-dimensional shift of finite type $(X, \mathbb Z)$ with entropy at least log n factors onto the full n-shift. The factor map is constructed by exploiting the fact that X, or a subshift of X, is conjugate to a shift of finite type in which every symbol can be followed by at least n symbols. We will investigate analogous statements for higher-dimensional shifts of finite type. We will also show that for a certain class of mixing higher-dimensional shifts of finite type, sufficient entropy implies that $(X,\mathbb Z^d )$ is finitely equivalent to a shift of finite type that maps onto the full n-shift.

We define a notion of the Haagerup property for measure-preserving standard equivalence relations. Given such a relation R on X with finite invariant measure $\mu$, we prove that R has the Haagerup property if and only if the associated finite von Neumann algebra L(R) (see J. Feldman and C. C. Moore. Ergodic equivalence relations, cohomology and von Neumann algebras II. Trans. Amer. Math. Soc.234 (1977), 325–350) has relative property H in the sense of Popa with respect to its natural Cartan subalgebra $L^{\infty}(X,\mu)$. We also prove that if G is a countable group such that R = RG has the Haagerup property and if R is ergodic, then G cannot have Kazhdan's property T.

The system $-d^2y/dt^2+u(t)y=Ey\ (y\in R^n)$ is considered, where $E=\text{diag} (\lambda_1^2,\, \lambda_2^2,\dots,\lambda_n^2)$ is a diagonal matrix with $\lambda_j>0\ (j=1,2,\dots, n)$ being regarded as parameters, and u a real analytic quasi-periodic symmetric matrix. Suppose that the basic frequencies of u satisfy a Diophantine condition. Then, for most of sufficiently large $\lambda_j\ (j=1,2,\dots,n)$, the system has n pairs of conjugate quasi-periodic solutions.

We study the entropy of actions on function spaces with the focus on doubly stochastic operators on probability spaces and Markov operators on compact spaces. Using an axiomatic approach to entropy we prove that there is basically only one reasonable measure-theoretic entropy notion on doubly stochastic operators. By ‘reasonable’ we mean extending the Kolmogorov–Sinai entropy on measure-preserving transformations and satisfying some obvious continuity conditions for $H_\mu$. In particular, this establishes equality on such operators between the entropy notion introduced by R. Alicki, J. Andries, M. Fannes and P. Tuyls (a version of which was also studied by I. I. Makarov), another notion of entropy introduced by E. Ghys, R. Langevin and P. Walczak, and our new definition introduced later in this paper. The key tool in proving this uniqueness is the discovery of a very general property of all doubly stochastic operators, which we call asymptotic lattice stability. Unlike the other explicit definitions of entropy mentioned above, ours satisfies many natural requirements already on the level of the function $H_\mu$, and we prove that the limit defining $h_\mu$ exists. The proof uses an integral representation of a stochastic operator obtained many years ago by A. Iwanik. In the topological part of the paper we introduce three natural definitions of topological entropy for Markov operators on C(X). Then we prove that all three are equal. Finally, we establish the partial variational principle: the topological entropy of a Markov operator majorizes the measure-theoretic entropy of this operator with respect to any of its invariant probability measures.

In this paper we study topological cocycles for minimal homeomorphisms on a compact metric space. We introduce a notion of an essential range for topological cocycles with values in a locally compact group, and we show that this notion coincides with the well-known topological essential range if the group is abelian. We then define a regularity condition for cocycles and prove several results on the essential ranges and the orbit closures of the skew product of regular cocycles. Furthermore, we show that recurrent cocycles for a minimal rotation on a locally connected compact group are always regular, assuming that their ranges are in a nilpotent group, and then their essential ranges are almost connected.

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.

We investigate rigidity of measurable structure for higher-rank Abelian algebraic actions. In particular, we show that ergodic measures for these actions fiber over a zero entropy measure with Haar measures along the leaves. We deduce various rigidity theorems for isomorphisms and joinings as corollaries.

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.

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

Nous montrons que l'entropie topologique des applications méromorphes n'est pas un invariant biméromorphe. Cela fournit un contre-exemple à une conjecture de Friedland. Nous proposons une version raffinée de cette conjecture.

We will deal with two topics about normality of the iterates of holomorphic self-maps in compact varieties. First, for a holomorphic self-map f of an irreducible compact variety Z, we show that if at least one subsequence of $\{f^n\}_{n\ge0}$ converges uniformly, then the full sequence $\{f^n\}_{n\ge0}$ itself is a normal family and the full set of the limit maps is finite or has the structure of a compact commutative Lie group. Second, in the case when Z is non-singular, we deal with the dynamics on forward-invariant compact subsets outside the closures of the post-critical sets. We will describe the semi-repelling structure of the dynamics in terms of repelling points and neutral Fatou discs (center manifolds). In particular, in the case of holomorphic self-maps of projective spaces, we will obtain a stronger result.

The two-sided shift on the infinite tensor product of copies of the $n\times n$ matrix algebra has the so-called Rohlin property, which entails the one-cocycle property, useful in analyzing cocycle-conjugacy classes. In the case n = 2, the restriction of the shift to the gauge-invariant CAR algebra also has the one-cocycle property. We extend the latter result to an arbitrary $n\geq 2$. As a corollary it follows that the flow $\alpha$ on the Cuntz algebra $\mathcal{O}_n=C^*(s_0,s_1,\dotsc,s_{n-1})$ defined by $\alpha_t(s_j)=e^{ip_jt}s_j$ has the Rohlin property (for flows) if and only if $p_0,\dotsc,p_{n-1}$ generate $\mathbb R$ as a closed sub-semigroup. Note that then such flows are all cocycle-conjugate to each other.

We show that among three-interval exchange transformations there exists a dichotomy: T has minimal self-joinings whenever the associated subshift is linearly recurrent, and is rigid otherwise. We also build a family of simple rigid three-interval exchange transformations, which is a step towards an old question of Veech, and a family of rigid three-interval exchange transformations, which includes Katok's rank-one map.

We study here the action of subgroups of PSL$(2,\mathbb{R})$ on the space of harmonic functions on the unit disc bounded by a common constant, as well as the relationship this action has with the foliated Liouville problem. Given a foliation of a compact manifold by Riemannian leaves and a leafwise harmonic continuous function on the manifold, is the function leafwise constant? We give a number of positive results and also show a general class of examples for which the Liouville property does not hold. The connection between the Liouville property and the dynamics on the space of harmonic functions as well as general properties of this dynamical system are explored. It is shown among other properties that the $\mathbb{Z}$-action generated by hyperbolic or parabolic elements of PSL$(2,\mathbb{R})$ is chaotic.

We present several results for the periods of periodic points of sup-norm non-expansive maps. In particular, we show that the period of each periodic point of a sup-norm non-expansive map $f\colon M\to M$, where $M\subset \mathbb{R}^n$, is at most $\max_k\, 2^k \big(\begin{smallmatrix}n\\ k\end{smallmatrix}\big)$. This upper bound is smaller than 3n and improves the previously known bounds. Further, we consider a special class of sup-norm non-expansive maps, namely topical functions. For topical functions $f\colon\mathbb{R}^n\to\mathbb{R}^n$ Gunawardena and Sparrow have conjectured that the optimal upper bound for the periods of periodic points is $\big(\begin{smallmatrix}n\\ \lfloor n/2\rfloor\end{smallmatrix}\big)$. We give a proof of this conjecture. To obtain the results we use combinatorial and geometric arguments. In particular, we analyse the cardinality of anti-chains in certain partially ordered sets.

Nous démontrons l'égalité, pour les convexes divisibles, entre la régularité Hölder du bord du convexe et la régularité Hölder du bord du convexe dual.

Let R be a rational map. A critical point c is called summable if the series $\sum_i(1/(R^i)'(R(c)))$ is absolutely convergent. Under certain topological conditions on the postcritical set we prove that R cannot be structurally stable if it has a summable critical point $c \in J(R)$.

We show that the orbit of a point on a compact nilmanifold X under the action of a polynomial sequence of translations on X is well distributed on the union of several sub-nilmanifolds of X. This implies that the ergodic averages of a continuous function on X along a polynomial sequence of translations on X converge pointwise.

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.