Ergodic Theory and Dynamical Systems最新文献

Based on previous work of the authors, to any S-adic development of a subshift X a ‘directive sequence’ of commutative diagrams is associated, which consists at every level $n geq 0$ of the measure cone and the letter frequency cone of the level subshift $X_n$ associated canonically to the given S-adic development. The issuing rich picture enables one to deduce results about X with unexpected directness. For instance, we exhibit a large class of minimal subshifts with entropy zero that all have infinitely many ergodic probability measures. As a side result, we also exhibit, for any integer $d geq 2$, an S-adic development of a minimal, aperiodic, uniquely ergodic subshift X, where all level alphabets $mathcal A_n$ have cardinality $d,$ while none of the $d-2$ bottom level morphisms is recognizable in its level subshift $X_n subseteq mathcal A_n^{mathbb {Z}}$.

The problem of non-integrability of the circular restricted three-body problem is very classical and important in the theory of dynamical systems. It was partially solved by Poincaré in the nineteenth century: he showed that there exists no real-analytic first integral which depends analytically on the mass ratio of the second body to the total and is functionally independent of the Hamiltonian. When the mass of the second body becomes zero, the restricted three-body problem reduces to the two-body Kepler problem. We prove the non-integrability of the restricted three-body problem both in the planar and spatial cases for any non-zero mass of the second body. Our basic tool of the proofs is a technique developed here for determining whether perturbations of integrable systems which may be non-Hamiltonian are not meromorphically integrable near resonant periodic orbits such that the first integrals and commutative vector fields also depend meromorphically on the perturbation parameter. The technique is based on generalized versions due to Ayoul and Zung of the Morales–Ramis and Morales–Ramis–Simó theories. We emphasize that our results are not just applications of the theories.

For a continuous $mathbb {N}^d$ or $mathbb {Z}^d$ action on a compact space, we introduce the notion of Bohr chaoticity, which is an invariant of topological conjugacy and which is proved stronger than having positive entropy. We prove that all principal algebraic $mathbb {Z}$ actions of positive entropy are Bohr chaotic. The same is proved for principal algebraic actions of $mathbb {Z}^d$ with positive entropy under the condition of existence of summable homoclinic points.

We describe two kinds of regular invariant measures on the boundary path space $partial E$ of a second countable topological graph E, which allows us to describe all extremal tracial weights on $C^{*}(E)$ which are not gauge-invariant. Using this description, we prove that all tracial weights on the C$^{*}$-algebra $C^{*}(E)$ of a second countable topological graph E are gauge-invariant when E is free. This in particular implies that all tracial weights on $C^{*}(E)$ are gauge-invariant when $C^{*}(E)$ is simple and separable.

Using Patterson–Sullivan measures, we investigate growth problems for groups acting on a metric space with a strongly contracting element.

We obtain the following embedding theorem for symbolic dynamical systems. Let G be a countable amenable group with the comparison property. Let X be a strongly aperiodic subshift over G. Let Y be a strongly irreducible shift of finite type over G that has no global period, meaning that the shift action is faithful on Y. If the topological entropy of X is strictly less than that of Y and Y contains at least one factor of X, then X embeds into Y. This result partially extends the classical result of Krieger when $G = mathbb {Z}$ and the results of Lightwood when $G = mathbb {Z}^d$ for $d geq 2$. The proof relies on recent developments in the theory of tilings and quasi-tilings of amenable groups.

Let G be a countable residually finite group (for instance, ${mathbb F}_2$) and let $overleftarrow {G}$ be a totally disconnected metric compactification of G equipped with the action of G by left multiplication. For every $rgeq 1$, we construct a Toeplitz G-subshift $(X,sigma ,G)$, which is an almost one-to-one extension of $overleftarrow {G}$, having r ergodic measures $nu _1, ldots ,nu _r$ such that for every $1leq ileq r$, the measure-theoretic dynamical system $(X,sigma ,G,nu _i)$ is isomorphic to