We characterize when C0-semigroups induced by semiflows are hypercyclic, topologically mixing, or chaotic both on spaces of integrable functions and on spaces of continuous functions. Furthermore, we give characterizations of transitivity for weighted composition operators on these spaces.

Given a compact surface $F$ with non-empty boundary and a homotopy class of self-maps on $F$, we consider the problem of finding a representative which carries the minimal number of fixed points possible for the class. When $F$ is either the pants surface or the once punctured Möbius band this problem is solved by a graph map, where the graph is homotopy equivalent to $F$. A partial result is presented for the remaining two surfaces whose fundamental group is the free group on two generators.

We prove the existence of $C^{1}$ critical subsolutions of the Hamilton–Jacobi equation for a time-periodic Hamiltonian system. We draw a consequence for the Minimal Action functional of the system.

An algebraic $\mathbb{Z}^d$-action of entropy rank one is one for which each element has finite entropy. Using the structure theory of these actions due to Einsiedler and Lind, this paper investigates dynamical zeta functions for elements of the action. An explicit periodic point formula is obtained leading to a uniform parameterization of the zeta functions that arise in expansive components of an expansive action, together with necessary and sufficient conditions for rationality in a more general setting.

We completely classify the Kubo–Martin–Schwinger (KMS) states for the gauge action on a C*-algebra associated with a rational function R introduced in our previous work. The gauge action has a phase transition at β=log deg R. We can recover the degree of R, the number of branched points, the number of exceptional points and the orbits of exceptional points from the structure of the KMS states. We also classify the KMS states for C*-algebras associated with some self-similar sets, including the full tent map and the Sierpinski gasket by a similar method.

Let ${\mathcal{A}}^{{\mathbb{Z}}^D}$ be the Cantor space of ${\mathbb{Z}}^D$-indexed configurations in a finite alphabet ${\mathcal{A}}$, and let $\sigma$ be the ${\mathbb{Z}}^D$-action of shifts on ${\mathcal{A}}^{{\mathbb{Z}}^D}$. A cellular automaton is a continuous, $\sigma$-commuting self-map $\Phi$ of ${\mathcal{A}}^{{\mathbb{Z}}^D}$, and a $\Phi$-invariant subshift is a closed, $(\Phi,\sigma)$-invariant subset ${\mathfrak{A}}\subset{\mathcal{A}}^{{\mathbb{Z}}^D}$. Suppose that ${\mathbf{a}}\in{\mathcal{A}}^{{\mathbb{Z}}^D}$ is ${\mathfrak{A}}$-admissible everywhere except for some small region that we call a defect. It has been empirically observed that such defects persist under iteration of $\Phi$, and often propagate like ‘particles’ which coalesce or annihilate on contact. We construct algebraic invariants for these defects, which explain their persistence under $\Phi$, and partly explain the outcomes of their collisions. Some invariants are based on the cocycles of multidimensional subshifts; others arise from the higher-dimensional (co)homology/homotopy groups for subshifts, obtained by generalizing the Conway–Lagarias tiling groups and the Geller–Propp fundamental group.

We generalize a series of topological Wiener–Wintner ergodic theorems due to Walters to the context of group extensions of measure-preserving transformations where the group is a non-abelian compact Lie group. Applications to random ergodic theorems for a shift map are given.

We generalize the classical result of J. Mather stating the existence of a drift orbit inside a region of instability of an exact-symplectic positive twist map, to the case of a finite family ${\cal F}$ of such maps. A special case is the case where the maps $F\in{\cal F}$ have no common invariant continuous graph. We prove the existence of a sequence $(z_i)_{i\in\mathbb{Z}}$ in $\mathbb{R}/\mathbb{Z}\times\mathbb{R}$ and of a sequence $(F_i)_{i\in\mathbb{Z}}$ in ${\cal F}$ such that

\[ z_{i+1}=F_i(z_i),\quad \lim_{i\to -\infty} p_2(z_i)=-\infty,\quad \lim_{i\to +\infty} p_2(z_i)=+\infty, \]

where $p_{2} :(x,y)\mapsto y$ is the second projection.

Let $X$ be a compact space, $f\colon X \to X$ a continuous map, and $\Lambda \subset X$ be any $f$-invariant subset. Assume that there exists a nested family of subsets $\{\Lambda_l\}_{l \geq 1}$ that exhaust $\Lambda$, that is $\Lambda_l \subset\Lambda_{l+1}$ and $\Lambda =\bigcup_{l \geq 1} \Lambda_l$. Assume that the potential $\varphi \colon X \to \mathbb{R}$ is continuous on the closure of each $\Lambda_l$ but not necessarily continuous on $\Lambda$. We define the topological pressure of $\varphi$ on $\Lambda$. This definition is shown to have a corresponding variational principle. We apply the topological pressure and variational principle to systems with non-zero Lyapunov exponents, countable Markov shifts, and unimodal maps.

We prove that the Collet–Eckmann condition for recurrent critical orbits inside the Julia set of a rational map with no parabolic periodic orbits implies uniform hyperbolicity on periodic orbits.

We give a surface for which the Ricci flow applied to the metric will increase the topological entropy of the geodesic flow. Specifically, we first adapt the Melnikov method to apply to a Ricci flow perturbation and then we construct a surface which is closely related to a surface of revolution, but does not quite have rotational symmetry. This is done by adapting the Liouville metric representation of a surface of revolution. The final steps of the Melnikov method require numerical integration.

We prove that if $G$ is a countable, discrete group having infinite normal subgroups with the relative property (T) of Kazhdan–Margulis, then the Bernoulli shift action of $G$ on $\prod_{g \in G} (X_0, \mu_0)_g$, for $(X_{0},\mu_{0})$ an arbitrary non-trivial probability space, has first cohomology group isomorphic to the character group of $G$.

Alves, Bonatti and Viana have recently shown that physical (or Sinai–Ruelle–Bowen) invariant measures exist for partially hyperbolic diffeomorphisms with mostly expanding center-unstable direction. In this paper we prove that such systems are statistically stable, that is, nearby diffeomorphisms have nearby physical measures.

A syzygy in the three-body problem is a collinear instant. We prove that, with the exception of Lagrange’s solution, every solution to the zero angular momentum, Newtonian three-body problem suffers syzygies. The proof works for all mass ratios.

In a previous paper (Presentations of subshifts and their topological conjugacy invariants. Doc. Math.4 (1999), 285–340), the notion of $\lambda$-graph system has been introduced. The $\lambda$-graph systems are generalizations of finite directed labeled graphs. In this paper, we study periodic points of $\lambda$-graph systems. We introduce some invariants for a $\lambda$-graph system $\mathfrak L$ to count the cardinal number of $p$-periodic points of $\mathfrak L$. They are invariant under strong shift equivalence of $\lambda$-graph systems. We then consider the zeta functions of $\lambda$-graph systems, which are also invariant under strong shift equivalence of $\lambda$-graph systems. Some examples are also presented.

Any continuous, transitive, piecewise monotonic map is determined up to a binary choice by its dimension module with the associated finite sequence of generators. The dimension module by itself determines the topological entropy of any transitive piecewise monotonic map, and determines any transitive unimodal map up to conjugacy. For a transitive piecewise monotonic map which is not essentially injective, the associated dimension group is a direct sum of simple dimension groups, each with a unique state.

Random diffeomorphisms with bounded absolutely continuous noise are known to possess a finite number of stationary measures. We discuss the dependence of stationary measures on an auxiliary parameter, thus describing bifurcations of families of random diffeomorphisms. A bifurcation theory is developed under mild regularity assumptions on the diffeomorphisms and the noise distribution (e.g. smooth diffeomorphisms with uniformly distributed additive noise are included). We distinguish bifurcations where the density function of a stationary measure varies discontinuously or where the support of a stationary measure varies discontinuously. We establish that generic random diffeomorphisms are stable. The densities of stable stationary measures are shown to be smooth and to depend smoothly on an auxiliary parameter, except at bifurcation values. The bifurcation theory explains the occurrence of transients and intermittency as the main bifurcation phenomena in random diffeomorphisms. Quantitative descriptions by means of average escape times from sets as functions of the parameter are provided. Further quantitative properties are described through the speed of decay of correlations as a function of the parameter. Random differentiable maps which are not necessarily injective are studied in one dimension; we show that stable one-dimensional random maps occur open and dense and that in one-parameter families bifurcations are typically isolated. We classify codimension-one bifurcations for one-dimensional random maps; we distinguish three possible kinds, the random saddle node, the random homoclinic and the random boundary bifurcation. The theory is illustrated on families of random circle diffeomorphisms and random unimodal maps.

We introduce and study a non-uniform hyperbolicity condition for complex rational maps that does not involve a growth condition. We call this condition backward contraction. We show this condition is weaker than the Collet–Eckmann condition, and than the summability condition with exponent one. Our main result is a connecting lemma for backward-contracting rational maps, roughly saying that we can perturb a rational map to connect each critical orbit in the Julia set with an orbit that does not accumulate on critical points. The proof of this result is based on Thurston's algorithm and some rigidity properties of quasi-conformal maps. We also prove that the Lebesgue measure of the Julia set of a backward-contracting rational map is zero, when it is not the whole Riemann sphere. The basic tool of this article is sets having a Markov property for backward iterates that are holomorphic analogues of nice intervals in real one-dimensional dynamics.

We show that the well-known figure-eight orbit of the three-body problem is linearly stable. Building on the strong amount of symmetry present, the monodromy matrix for the figure-eight is factored so that its stability can be determined from the first twelfth of the orbit. Using a clever change of coordinates, the problem is then reduced to a 2×2 matrix whose entries depend on solutions of the associated linear differential system. These entries are estimated rigorously using only a few steps of a Runge–Kutta–Fehlberg algorithm. From this, we conclude that the characteristic multipliers are distinct and lie on the unit circle. The methods and results presented are applicable to a wide range of Hamiltonian systems containing symmetric periodic solutions.

In the 1970s, Nekhorochev proved that, for an analytic nearly integrable Hamiltonian system, the action variables of the unperturbed Hamiltonian remain nearly constant over an exponentially long time with respect to the size of the perturbation, provided that the unperturbed Hamiltonian satisfies some generic transversality condition known as steepness. Recently, Guzzo has given examples of exponentially stable integrable Hamiltonians that are non-steep but satisfy a weak condition of transversality which involves only the affine subspaces spanned by integer vectors. We generalize this notion for an arbitrary integrable Hamiltonian and prove Nekhorochev's estimates in this setting. The point in this refinement lies in the fact that it allows one to exhibit a generic class of real analytic integrable Hamiltonians which are exponentially stable with fixed exponents. Genericity is proved in the sense of measure since we exhibit a prevalent set of integrable Hamiltonians which satisfy the latter property. This is obtained by an application of a quantitative Sard theorem given by Yomdin.