For projectivizations of rational maps, Bellon and Viallet defined the notion of algebraic entropy using the exponential growth rate of the degrees of iterates. We want to call this notion to the attention of dynamicists by computing algebraic entropy for certain rational maps of projective spaces (Theorem 6.2) and comparing it with topological entropy (Theorem 5.1). The particular rational maps we study are monomial maps (Definition 1.2), which are closely related to toral endomorphisms. Theorems 5.1 and 6.2 that imply that the algebraic entropy of a monomial map is always bounded above by its topological entropy, and that the inequality is strict if the defining matrix has more than one eigenvalue outside the unit circle. Also, Bellon and Viallet conjectured that the algebraic entropy of every rational map is the logarithm of an algebraic integer, and Theorem 6.2 establishes this for monomial maps. However, a simple example using a monomial map shows that a stronger conjecture of Bellon and Viallet is incorrect, in that the sequence of algebraic degrees of the iterates of a rational map of projective space need not satisfy a linear recurrence relation with constant coefficients.

A sufficient condition for unique ergodicity of symbolic flows is provided. In an important but special case of interval exchange transformations, the condition has already been validated by W. Veech.

For a sequence $(c_n)$ of complex numbers we consider the quadratic polynomials $f_{c_n}(z):=z^2+c_n$ and the sequence $(F_n)$ of iterates $F_n:= f_{c_n} \circ \dotsb \circ f_{c_1}$. The Fatou set $\mathcal{F}_{(c_n)}$ is by definition the set of all $z \in \widehat{\mathbb{C}}$ such that $(F_n)$ is normal in some neighbourhood of $z$, while the complement of $\mathcal{F}_{(c_n)}$ is called the Julia set $\mathcal{J}_{(c_n)}$. The aim of this paper is to study the connectedness of the Julia set $\mathcal{J}_{(c_n)}$ provided that the sequence $(c_n)$ is bounded and randomly chosen. For example, we prove a necessary and sufficient condition for the connectedness of $\mathcal{J}_{(c_n)}$ which implies that $\mathcal{J}_{(c_n)}$ is connected if $|c_n| \le \frac{1}{4}$, while it is almost surely disconnected if $|c_n| \le \delta$ for some $\delta>\frac{1}{4}$.

A Hamiltonian system of n degrees of freedom, defined by the function F, with an equilibrium point at the origin, is called formally integrable if there exist A A formal power series , functionally independent, in involution, and such that the Taylor expansion of F is a formal power series in the .

Take n = 3, , F(k) homogeneous of degree k, F(2) > 0 and the eigenfrequencies in ratio 1:1:2. If F(3) avoids a certain hypersurface of ‘symmetric’ third order terms, then the F system is not formally integrable. If F(3) is symmetric but F(4) is in a non-void open subset, then homoclinic intersection with Devaney spiralling occurs; the angle decays of order 1 when approaching the origin.

This paper uses an elementary variational argument to establish the existence of solutions heteroclinic to a pair of periodic orbits for a class of Hamiltonian systems including Hamiltonians of multiple pendulum type.

It is shown that any group isomorphism between the Ck diffeomorphism groups of two finite-dimensional, boundaryless paracompact manifolds is induced by a Ck diffeomorphism between the manifolds (1 ≤ k ≤ ∞).

In this paper we make precise the relationship between local or pointwise dimension and the dimension structure of Borel probability measures on metric spaces. Sufficient conditions for exact-dimensionality of the stationary ergodic distributions associated with a dynamical system are obtained. A counterexample is provided to show that ergodicity alone is not sufficient to guarantee exactdimensionality even in the case of continuous maps or flows.

We survey a collection of results in the dimension theory of dynamical systems, with emphasis on the study of repellers and hyperbolic sets of smooth dynamics. We discuss the most preeminent results in the area as well as the main difficulties in developing a general theory. Despite many interesting and non-trivial developments, only the case of conformal dynamics is completely understood. The study of the dimension of invariant sets of non-conformal maps has unveiled several new phenomena, but it still lacks today a satisfactory general approach. Indeed, we have a complete understanding of only a few classes of invariant sets of non-conformal maps satisfying certain simplifying assumptions. For example, the assumptions may ensure that there is a clear separation between different Lyapunov directions or that number-theoretical properties do not influence the dimension.

Hölder continuous observations on hyperbolic basic sets satisfy strong statistical properties such as exponential decay of correlations, central limit theorems and invariance principles (approximation by Brownian motion).

Using an equivariant version of the Ruelle transfer operator studied by Parry and Pollicott, we obtain similar results for equivariant observations on compact group extensions of hyperbolic basic sets.

For L equal to the unilateral or bilateral shift on a weighted sequence space, we characterize, in terms of f and the weight function, those f holomorphic on the spectrum of L for which f(L) is a chaotic operator. For B equal to d/dx, the generator of left translation, on weighted L^p spaces on [0, \infty) or \mathbb{R}, we similarly characterize those polynomials Q for which the differential operator Q(B) generates a chaotic semigroup.

The representation of Cantor minimal systems by Bratteli–Vershik diagrams has been extensively used to study particular aspects of their dynamics. A main role has been played by the symbolic factors induced by the way vertices of a fixed level of the diagram are visited by the dynamics. The main result of this paper states that Cantor minimal systems that can be represented by Bratteli–Vershik diagrams with a uniformly bounded number of vertices at each level (called finite-rank systems) are either expansive or topologically conjugate to an odometer. More precisely, when expansive, they are topologically conjugate to one of their symbolic factors.

The global maps of homomorphisms of directed graphs are very closely related to homomorphisms of a class of symbolic dynamical systems called subshifts of finite type. In this paper, we introduce the concepts of ‘induced regular homomorphism’ and ‘induced backward regular homomorphism’ which are associated with every homomorphism between strongly connected graphs whose global map is finite-to-one and onto, and using them we study the structure of constant-to-one and onto global maps of homorphisms between strongly connected graphs and that of constant-to-one and onto homomorphisms of irreducible subshifts of finite type. We determine constructively, up to topological conjugacy, the subshifts of finite type which are constant-to-one extensions of a given irreducible subshift of finite type. We give an invariant for constant-to-one and onto homomorphisms of irreducible subshifts of finite type.

Under a natural assumption the Hausdorff dimension of a measure μ canonically associated with a given self-affine set is computed. A simplified proof of Bowen's formula for the box dimension of self-affine sets proved earlier is given. A condition for the box dimension and Hausdorff dimension to be equal is proven, and a collection of examples in which this condition can be checked is discussed.

We show that equilibrium states μ of a function φ on ([0,1], T), where T is piecewise monotonic, have strong ergodic properties in the following three cases:

(i) sup φ — inf φ <htop(T) and φ is of bounded variation.

(ii) φ satisfies a variation condition and T has a local specification property.

(iii) φ = —log |T′|, which gives an absolutely continuous μ, T is C2, the orbits of the critical points of T are finite, and all periodic orbits of T are uniformly repelling.

In this paper we prove that the Poincaré map associated to a Lorenz-like flow has exponential decay of correlations with respect to Lipschitz observables. This implies that the hitting time associated to the flow satisfies a logarithm law. The hitting time τr(x,x0) is the time needed for the orbit of a point x to enter a ball Br(x0) centered at x0, with small radius r, for the first time. As the radius of the ball decreases to 0 its asymptotic behavior is a power law whose exponent is related to the local dimension of the physical measure at x0: for each x0 such that the local dimension dμ (x0) exists, holds for μ almost each x. In a similar way, it is possible to consider a quantitative recurrence indicator quantifying the speed of an orbit to come back to its starting point. Similar results hold for this recurrence indicator.

This paper is about spectral properties of transfer operators for contact Anosov flows. The main result gives the essential spectral radii of the transfer operators acting on an appropriate function space exactly and improves the previous result in Tsujii [Quasi-compactness of transfer operators for contact Anosov flows. Nonlinearity23 (2010), 1495–1545]. Also, we provide a simplified proof by using the so-called Fourier–Bros–Iagolnitzer (FBI) (or Bargmann) transform.

We consider a random walk on the affine group of the real line, we denote by P the corresponding Markov operator on , and we study the Birkhoff sums associated with its trajectories. We show that, depending on the parameters of the random walk, the normalized Birkhoff sums converge in law to a stable law of exponent α∈ ]0,2[ or to a normal law. The corresponding analysis is based on the spectral properties of two families of associated transfer operators Pt,Tt. The operator Pt is a Fourier operator and is considered here as a perturbation of the Markov operator P=P0 of the random walk. The operator Tt is related to Pt by a symmetry of Heisenberg type and is also considered as a perturbation of the Markov operator T0=T. We prove that these operators have an isolated dominant eigenvalue which has an asymptotic expansion involving fractional powers of t. The parameters of this expansion have simple expressions in terms of tails and moments of the stationary measures of P and T.

Using pressure formulas we compute the Hausdorff dimension of the basic set of ‘almost every’ ${\cal C}^{1+\alpha}$ horseshoe map in $\mathbb{R}}^3$ of the form $F(x,y,z)= (\gamma(x,z), \tau(y,z), \psi(z))$, where $|\psi'|>1$ and $0< |\gamma'_x|, |\tau'_y| < \frac{1}{2}$ on the basic set. Similar results are obtained for attractors of nonlinear ‘baker's maps’ in $\mathbb{R}}^3$.

We introduce two properties of dynamical systems on Polish metric spaces: closeability and linkability. We show that they imply density of ergodic measures in the space of invariant probability measures and the existence of a generic point for every invariant measure. In the compact case, it follows from our conditions that the set of invariant measures is either a singleton of a measure concentrated on a periodic orbit or the Poulsen simplex. We provide examples showing that closability and linkability are independent properties. Our theory applies to systems with the periodic specification property, irreducible Markov chains over a countable alphabet, certain coded systems including $\unicode[STIX]{x1D6FD}$-shifts and $S$-gap shifts, $C^{1}$-generic diffeomorphisms of a compact manifold $M$ and certain geodesic flows of a complete connected negatively curved manifold.

This self-contained paper is part of a series (Groups of homeomorphisms of one-manifolds I: actions of nonlinear groups. Preprint, 2001; Group actions on one-manifolds II: extensions of Hölder's Theorem. To appear in Trans. Amer. Math. Soc.) seeking to understand groups of homeomorphisms of manifolds in analogy with the theory of Lie groups and their discrete subgroups. Plante and Thurston proved that every nilpotent subgroup of Diff2(S1) is abelian. One of our main results is a sharp converse: Diff1(S1) contains every finitely generated, torsion-free nilpotent group.