A sequence of non-negative integers is said to be realizable if there is a map T of a set X such that ϕn=#{x:Tnx=x}. We prove that any realizable sequence can be realized by a diffeomorphism of .

We provide a complete system of invariants for the formal classification of unfoldings φ(x,x1,…,xn)=(f(x,x1,…,xn),x1,…,xn) of complex analytic germs of diffeomorphisms at that are tangent to the identity. We reduce the formal classification problem to solving a linear differential equation. Then we describe the formal invariants; their nature depends on the position of the fixed points set Fix φ with respect to the regular vector field ∂/∂x. We get invariants specifically attached to higher dimension (n≥3), although generically they are analogous to the one-dimensional ones.

Given a Polish space X, a countable Borel equivalence relation E on X, and a Borel cocycle , we characterize the circumstances under which there is a probability measure μ on X such that ρ(ϕ−1(x),x)=[d(ϕ*μ)/dμ](x) μ-almost everywhere, for every Borel injection ϕ whose graph is contained in E.

Let the vector field on M define a flow (fat) with an Axiom A attractor Λa depending continuously on a∈(−ϵ,ϵ). Let ρa be the SRB measure on Λa for (fat). If , then is on (−ϵ,ϵ) and dρa(A)/da is the limit when ω→0 with Im ω>0 of

Let M be a positive integer and let f be a holomorphic mapping from a ball Δn={x∈ℂn:∣x∣<δ} into ℂn such that the origin 0 is an isolated fixed point of both f and fM, the Mth iteration of f. Then one can define the number 𝒪M(f,0), interpreted as the number of periodic orbits of f with period M that are hidden at the fixed point 0. For an n×n matrix A whose eigenvalues are all the same primitive Mth root of unity, we give a sufficient and necessary condition on A such that for any holomorphic mapping f:Δn→ℂn with f(0)=0 and Df(0)=A, if 0 is an isolated fixed point of the Mth iteration fM , then 𝒪M (f,0)≥2.

We give a correction to the assumption of Theorems 1.12 and 2.6 in the paper [H. Sumi. Semi-hyperbolic fibered rational maps and rational semigroups. Ergod. Th. & Dynam. Sys.26 (2006), 893–922].

In this paper we consider a special class of polymorphisms with invariant measure, the algebraic polymorphisms of compact groups. A general polymorphism is—by definition—a many-valued map with invariant measure, and the conjugate operator of a polymorphism is a Markov operator (i.e. a positive operator on L2 of norm 1 which preserves the constants). In the algebraic case a polymorphism is a correspondence in the sense of algebraic geometry, but here we investigate it from a dynamical point of view. The most important examples are the algebraic polymorphisms of a torus, where we introduce a parametrization of the semigroup of toral polymorphisms in terms of rational matrices and describe the spectra of the corresponding Markov operators. A toral polymorphism is an automorphism of if and only if the associated rational matrix lies in . We characterize toral polymorphisms which are factors of toral automorphisms.

Let N be an integer with N≥2 and let X be a compact subset of ℝd. If is a list of contracting similarities Si:X→X, then we will write for the self-similar set associated with , and we will write M for the family of all lists satisfying the strong separation condition. In this paper we show that the maps (1)and (2)are continuous; here denotes the Hausdorff dimension, ℋs denotes the s-dimensional Hausdorff measure and 𝒮s denotes the s-dimensional spherical Hausdorff measure. In fact, we prove a more general continuity result which, amongst other things, implies that the maps in (1) and (2) are continuous.

In this paper we prove a renewal-type limit theorem. Given and R>0, let qnR be the first denominator of the convergents of α which exceeds R. The main result in the paper is that the ratio qnR/R has a limiting distribution as R tends to infinity. The existence of the limiting distribution uses mixing of a special flow over the natural extension of the Gauss map.

A class of discrete dynamical systems called projectively (or geodesically) equivalent Lagrangian systems is defined. We prove that these systems admit families of integrals. In the case of geodesically equivalent billiard tables, these integrals are pairwise commuting. We describe a family of geodesically equivalent billiard tables on surfaces of constant curvature. This is a special case of the so-called ‘Liouville billiard tables’.

Let T1,…,Tl:X→X be commuting measure-preserving transformations on a probability space . We show that the multiple ergodic averages are convergent in as for all ; this was previously established for l=2 by Conze and Lesigne [J. P. Conze and E. Lesigne. Théorèmes ergodique por les mesures diagonales. Bull. Soc. Math. France112 (1984), 143–175] and for general l assuming some additional ergodicity hypotheses on the maps Ti and TiTj−1 by Frantzikinakis and Kra [N. Frantzikinakis and B. Kra. Convergence of multiple ergodic averages for some commuting transformations. Ergod. Th. & Dynam. Sys.25 (2005), 799–809] (with the l=3 case of this result established earlier by Zhang [Q. Zhang. On the convergence of the averages . Mh. Math.122 (1996), 275–300]). Our approach is combinatorial and finitary in nature, inspired by recent developments regarding the hypergraph regularity and removal lemmas, although we will not need the full strength of those lemmas. In particular, the l=2 case of our arguments is a finitary analogue of those by Conze and Lesigne.

We show that every ergodic, finite entropy transformation which satisfies the weak Pinsker property possesses a finite generator whose two-sided tail field is exactly the Pinsker algebra. This is proved by exhibiting a generator endowed with a block structure quite analogous to the one appearing in the construction of the Ornstein–Shields examples of non Bernoulli K-automorphisms. We also show that, given two transformations T1 and T2 in the previous class (i.e. satisfying the weak Pinsker property), and a Bernoulli shift B, if T1×B is isomorphic to T2×B, then T1 is isomorphic to T2. That is, one can ‘factor out’ Bernoulli shifts.

In general, the entropy of a stationary process , which is a factor of the process , can take any value in the interval . For linear factors the situation is completely different. In fact it even matters how the class of linear factors is defined. We will investigate both moving averages of bounded sequences by coefficients in l1 and the natural notion in L2.