We obtain sharp estimates on the growth rate of stable commutator length on random (geodesic) words, and on random walks, in hyperbolic groups and groups acting non-degenerately on hyperbolic spaces. In either case, we show that with high probability stable commutator length of an element of length $n$ is of order $n/ \log n$. This establishes quantitative refinements of qualitative results of Bestvina and Fujiwara and others on the infinite dimensionality of two-dimensional bounded cohomology in groups acting suitably on hyperbolic spaces, in the sense that we can control the geometry of the unit balls in these normed vector spaces (or rather, in random subspaces of their normed duals). As a corollary of our methods, we show that an element obtained by random walk of length $n$ in a mapping class group cannot be written as a product of fewer than $O(n/ \log n)$ reducible elements, with probability going to $1$ as $n$ goes to infinity. We also show that the translation length on the complex of free factors of a random walk of length $n$ on the outer automorphism group of a free group grows linearly in $n$.

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.

We establish a generalized thermodynamic formalism for certain nonhyperbolic maps with countably many preimages. We study existence and uniqueness of conformal measures and statistical properties of the equilibrium states absolutely continuous with respect to the conformal measures. We will see that such measures are not Gibbs but satisfy a version of Gibbs property (weak Gibbs measure). We apply our results to a one-parameter family of one-dimensional maps and a two-dimensional nonconformal map related to number theory. Both of them admit indifferent periodic points.

The equilateral triangle family of relative equilibria of the 4-body problem consists of three particles of mass 1 at the vertices of an equilateral triangle and the fourth particle of arbitrary mass m at the centroid. For one value of the mass m this relative equilibrium is degenerate. We show that families of isosceles triangle relative equilibria bifurcate from the equilateral triangle family as m passes through the degenerate value.

The square family of relative equilibria of the 5-body problem consists of four particles of mass 1 at the vertices of a square and the fifth particle of arbitrary mass m at the centroid. For one value of the mass m this relative equilibrium is degenerate. We show that families of kite and isosceles trapezoidal relative equilibria bifurcate from the square family as m passes through the degenerate value.

Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ be a homeomorphism of the plane. We define open sets $P \subset \mathbb{R}^{2}$, called pruning fronts after the work of Cvitanović et al. in 1988, for which it is possible to construct an isotopy $H: \mathbb{R}^{2} \times [0,1] \rightarrow \mathbb{R}^{2}$ with open support contained in $\bigcup _{n \in {\mathbb{Z}} } f^{n} (P)$ such that $H(\cdot, 0 ) = f(\cdot)$ and $H(\cdot, 1) = f_{P} (\cdot)$, where $f_P$ is a homeomorphism under which every point of $P$ is wandering. Applying this construction when $f$ is Smale's horseshoe, it is possible to obtain an uncountable family of homeomorphisms, depending on infinitely many parameters, going from trivial to chaotic dynamic behavior. This family is a two-dimensional analog of a one-dimensional universal family.

Let β∈(1,2). Each x∈[0,1/(β−1)] can be represented in the form where εk∈{0,1} for all k (a β-expansion of x). If , then, as is well known, there always exist x∈(0,1/(β−1)) which have a unique β-expansion. We study (purely) periodic unique β-expansions and show that for each n≥2 there exists such that there are no unique periodic β-expansions of smallest period n for β≤βn and at least one such expansion for β>βn. Furthermore, we prove that βk<βm if and only if k is less than m in the sense of the Sharkovskiĭ ordering. We give two proofs of this result, one of which is independent, and the other one links it to the dynamics of a family of trapezoidal maps.

Downarowicz [Entropy structure. J. Anal.96 (2005), 57–116] stated a variational principle for the tail entropy for invertible continuous dynamical systems of a compact metric space. We give here an elementary proof of this variational principle. Furthermore, we extend the result to the non-invertible case.

We compute the Minkowski dimension for a family of self-affine sets on ℝ2. Our result holds for every (rather than generic) set in the class. Moreover, we exhibit explicit open subsets of this class where we allow overlapping, and do not impose any conditions on the norms of the linear maps. The family under consideration was inspired by the theory of Kakeya sets.

Krieger's embedding theorem for $\mathbb{Z}$ mixing shifts for finite type (SFTs) is extended to the $\mathbb{Z}^2$ case. We prove that if X is a non-periodic subshift ($\sigma^vx=x\Rightarrow v=0\in\mathbb{Z}^2$) and Y is a $\mathbb{Z}^2$ square-filling mixing SFT, then there exists a homomorphism $X\to Y$. The proof is a construction which begins by constructing Voronoi tilings using techniques from Part I (S. Lightwood. Morphisms from non-periodic $\mathbb{Z}^2$ subshifts I: constructing embeddings from homomorphisms. Ergod. Th. & Dynam. Sys.23 (2003), 587–609.). A tiling by Delaunay polygons is derived from Voronoi tiling. The union of the boundaries of the Delaunay polygons (referred to as the Delaunay graph) is itself tiled by trees. Words painted on the thickened trees combine to form words on the thickened infinite Delaunay graph. It is the point and sole purpose of square-filling that words on such thickened infinite graphs will correspond to points in the target space.

Combined with the results from Part I, this gives us the $\mathbb{Z}^2$ extension of Krieger's Embedding Theorem: if X is a non-periodic subshift and Y is a $\mathbb{Z}^2$ square-filling mixing SFT, then there exists an embedding $X\hookrightarrow Y$ if and only if $h(X)<h(Y)$, where h denotes the $\mathbb{Z}^2$ entropy. The techniques developed here play a central role in the proof of an embedding theorem for general $\mathbb{Z}^2$ subshifts into square-filling mixing SFTs which will be carried out in a subsequent paper.

Bonatti and Viana introduced a robust (non-empty interior) class of partially hyperbolic attractors of C2-diffeomorphisms on a compact manifold, for which they construct Sinai–Ruelle–Bowen measures. For some such robust examples, we prove the exponential decay of correlations and the central limit theorem, in the space of Hölder continuous functions. For the proof, we adapt the techniques (backward inducing, redundancy elimination algorithm) we have previously developed.

Interval exchange maps are related to geodesic flows on translation surfaces; they correspond to the first return maps of the vertical flow on a transverse segment. The Rauzy–Veech induction on the space of interval exchange maps provides a powerful tool to analyze the Teichmüller geodesic flow on the moduli space of Abelian differentials. Several major results have been proved using this renormalization. Danthony and Nogueira introduced in 1988 a natural generalization of interval exchange transformations, namely linear involutions. These maps are related to general measured foliations on surfaces (whether orientable or not). In this paper we are interested by such maps related to geodesic flow on (orientable) flat surfaces with ℤ/2ℤ linear holonomy. We relate geometry and dynamics of such maps to the combinatorics of generalized permutations. We study an analogue of the Rauzy–Veech induction and give an efficient combinatorial characterization of its attractors. We establish a natural bijection between the extended Rauzy classes of generalized permutations and connected components of the strata of meromorphic quadratic differentials with at most simple poles, which allows us, in particular, to classify the connected components of all exceptional strata.

We consider in this note smooth dynamical systems equipped with smooth invariant affine connections and show that, under a pinching condition on the Lyapunov exponents, certain invariant tensor fields are parallel. We then apply this result to a problem of rigidity of geodesic flows for Riemannian manifolds with negative curvature.

Let V be a finite-dimensional vector space over $\mathbb{R}$and let $\Gamma \subset \mathrm{GL}(V)$ be a semigroup. We study the closed $\Gamma$-invariant subsets of V-\{0\} under the condition that the Zariski closure of $\Gamma$ is semi-simple. We use the results to show that, if $\Gamma\subset\mathrm{SL}(\mathbb{R}^d)$ acts on $\mathbb{T}^d$ by automorphisms, then the orbits of $\Gamma$ are finite or dense.

If is a weakly mixing isometric extension of a finite measure preserving, k-fold mixing map T, then must also be k-fold mixing.

We investigate the dynamics of polynomial semigroups (semigroups generated by a family of polynomial maps on the Riemann sphere ) and the random dynamics of polynomials on the Riemann sphere. Combining the dynamics of semigroups and the fiberwise (random) dynamics, we give a classification of polynomial semigroups G such that G is generated by a compact family Γ, the planar postcritical set of G is bounded, and G is (semi-) hyperbolic. In one of the classes, we have that, for almost every sequence , the Julia set Jγ of γ is a Jordan curve but not a quasicircle, the unbounded component of is a John domain, and the bounded component of is not a John domain. Note that this phenomenon does not hold in the usual iteration of a single polynomial. Moreover, we consider the dynamics of polynomial semigroups G such that the planar postcritical set of G is bounded and the Julia set is disconnected. Those phenomena of polynomial semigroups and random dynamics of polynomials that do not occur in the usual dynamics of polynomials are systematically investigated.

The paper deals with the variational principles for evaluation of the spectral radii of transfer and weighted shift operators associated with a dynamical system. These variational principles have been the matter of numerous investigations and the principal results have been achieved in the situation when the dynamical system is either reversible or a topological Markov chain. As the main summands, these principles contain the integrals over invariant measures and the Kolmogorov–Sinai entropy. In the paper we derive the variational principle for an arbitrary dynamical system. It gives the explicit description of the Legendre dual object to the spectral potential. It is shown that in general this principle contains not the Kolmogorov–Sinai entropy but a new invariant of entropy type—the t-entropy.

We find an explicit formula for the invariant density h of an arbitrary eventually expanding piecewise linear map τ of an interval [0,1]. We do not assume that the slopes of the branches are the same and we allow arbitrary number of shorter branches touching zero or touching one or hanging in between. The construction involves the matrix S which is defined in a way somewhat similar to the definition of the kneading matrix of a continuous piecewise monotonic map. Under some additional assumptions, we prove that if 1 is not an eigenvalue of S, then the dynamical system (τ,h⋅m) is ergodic with full support.

Two notions of metric invariance entropy are constructed with respect to conditionally invariant measures for control systems in discrete time and it is shown that they are invariant under conjugacies.

An interval map with only one discontinuity is isomorphic to a rotation of the circle, and has continuous eigenfunctions. What we show here is that for almost every choice of lengths of the intervals, this is the only way an irreducible interval exchange can have a somewhere continuous eigenfunction. We show slightly more, considering certain towers over the interval exchange, showing that outside of a set of choices for interval lengths of measure zero these have a somewhere continuous eigenfunction only if they are isomorphic to either a rotation, or a tower of constant height over an interval exchange.

We will consider the local dimension spectrum of a weak Gibbs measure on a C1 non-uniformly hyperbolic system of Manneville–Pomeau type. We will present the spectrum in three ways: using invariant measures, ergodic invariant measures supported on hyperbolic sets and equilibrium states. We are also proving analyticity of the spectrum under additional assumptions. All three presentations are well known for smooth uniformly hyperbolic systems.