Zero-dimensional and symbolic extensions of topological flows

<p style='text-indent:20px;'>A zero-dimensional (resp. symbolic) flow is a suspension flow over a zero-dimensional system (resp. a subshift). We show that any topological flow admits a principal extension by a zero-dimensional flow. Following [<xref ref-type="bibr" rid="b6">6</xref>] we deduce that any topological flow admits an extension by a symbolic flow if and only if its time-<inline-formula><tex-math id="M1">\begin{document}$ t $\end{document}</tex-math></inline-formula> map admits an extension by a subshift for any <inline-formula><tex-math id="M2">\begin{document}$ t\neq 0 $\end{document}</tex-math></inline-formula>. Moreover the existence of such an extension is preserved under orbit equivalence for regular topological flows, but this property does not hold for singular flows. Finally we investigate symbolic extensions for singular suspension flows. In particular, the suspension flow over the full shift on <inline-formula><tex-math id="M3">\begin{document}$ \{0,1\}^{\mathbb Z} $\end{document}</tex-math></inline-formula> with a roof function <inline-formula><tex-math id="M4">\begin{document}$ f $\end{document}</tex-math></inline-formula> vanishing at the zero sequence <inline-formula><tex-math id="M5">\begin{document}$ 0^\infty $\end{document}</tex-math></inline-formula> admits a principal symbolic extension or not depending on the smoothness of <inline-formula><tex-math id="M6">\begin{document}$ f $\end{document}</tex-math></inline-formula> at <inline-formula><tex-math id="M7">\begin{document}$ 0^\infty $\end{document}</tex-math></inline-formula>.</p>

Symbolic dynamics for non-uniformly hyperbolic systems

This survey describes the recent advances in the construction of Markov partitions for non-uniformly hyperbolic systems. One important feature of this development comes from a finer theory of non-uniformly hyperbolic systems, which we also describe. The Markov partition defines a symbolic extension that is finite-to-one and onto a non-uniformly hyperbolic locus, and this provides dynamical and statistical consequences such as estimates on the number of closed orbits and properties of equilibrium measures. The class of systems includes diffeomorphisms, flows, and maps with singularities.

Courtesan Singers as Courtiers: Power, Political Pawns, and the Arrest of virtuosa Nina Barcarola

This article gives a close reading of the “avvisi di Roma”—unpublished archival documents reporting on daily life in the city—that record the arrest in 1645 of famous Roman courtesan singer Nina Barcarola. Organized by the political enemies of Nina's main protector, Cardinal Antonio Barberini, the arrest was orchestrated so as to compromise the public honor of both. The reports of the arrest reflect a growing elite interest in female vocal performance in Rome, and attest to a rise in the social value of courtesan singers. Examining details provided in these reports, the article explores various aspects of Nina's life and courtesan singing culture more generally: the public honor and social practices of courtesan singers; the positive effect of singing on courtesan honor; the types of gatherings hosted by Nina; and her politically satirical public performances. It also analyzes Nina's relationship to various areas of contemporary politics—social, state, familial, and gender. The reports reveal that, in the public sphere, Nina, like Barberini's male dependents, served as a symbolic extension of the cardinal. By introducing courtesan singers—a significant, marginalized population—into musicological discourse on seventeenth-century Rome, the article broadens our understanding of Roman singing culture in this period.

HURRICANES, CROPS, AND CAPITAL: THE METEOROLOGICAL INFRASTRUCTURE OF AMERICAN EMPIRE IN THE WEST INDIES

This article examines the mutually reinforcing imperatives of government science, capitalism, and American empire through a history of the U.S. Weather Bureau's West Indian weather service at the turn of the twentieth century. The original impetus for expanding American meteorological infrastructure into the Caribbean in 1898 was to protect naval vessels from hurricanes, but what began as a measure of military security became, within a year, an instrument of economic expansion that extracted climatological data and produced agricultural reports for American investors. This article argues that the West Indian weather service was a project of imperial meteorology that sought to impose a rational scientific and bureaucratic order on a region that American officials considered racially and culturally inferior, yet relied on the labor of local observers and Cuban meteorological experts in order to do so. Weather reporting networks are examined as a material and symbolic extension of American technoscientific power into the Caribbean and as a knowledge infrastructure that linked the production of agricultural commodities in Cuba and Puerto Rico to the world of commodity exchange in the United States.

Modeling potential as fiber entropy and pressure as entropy

We first prove that topological fiber entropy potential in a relatively symbolic extension (topological joining with a subshift) of a topological dynamical system (which is a non-negative, non-decreasing, upper semicontinuous and subadditive potential$\mathfrak{H}$) yields topological pressure equal to the topological entropy of the extended system. The terms occurring on the other side of the variational principle for pressure are equal to the extension entropies of the invariant measures. Thus the variational principle for pressure reduces to the usual variational principle (for entropy) applied to the extended system. Next we prove our main theorem saying that every non-negative, upper semicontinuous and subadditive potential$\mathfrak{F}$(we drop the monotonicity assumption) is ‘nearly equal’ to the fiber entropy potentialHin some relatively symbolic extension of the system, in the sense that all terms occurring in the variational principle for pressure are the same for both potentials. This gives a new interpretation of all such potentials$\mathfrak{F}$as a kind of additional information function enhancing the natural information arising from the dynamical system, and provides a new proof of the variational principle for pressure. At the end of the paper we provide examples showing that both assumptions, continuity and additivity, under which so-called lower pressure (defined with the help of spanning sets) equals the pressure, are essential, already in the class of non-negative, upper semicontinuous, subadditive potentials.

A lower bound for topological entropy of generic non-Anosov symplectic diffeomorphisms

We prove that a${C}^{1} $generic symplectic diffeomorphism is either Anosov or its topological entropy is bounded from below by the supremum over the smallest positive Lyapunov exponent of its periodic points. We also prove that${C}^{1} $generic symplectic diffeomorphisms outside the Anosov ones do not admit symbolic extension and, finally, we give examples of volume preserving surface diffeomorphisms which are not points of upper semicontinuity of the entropy function in the${C}^{1} $topology.

Embedding ℤk-actions in cubical shifts and ℤk-symbolic extensions

Mean dimension is an invariant which makes it possible to distinguish between topological dynamical systems with infinite entropy. Extending in part the work of Lindenstrauss we show that if (X,ℤk) has a free zero-dimensional factor then it can be embedded in the ℤk-shift on ([0,1]d)ℤk, where d=[C(k) mdim(X,ℤk)]+1 for some universal constant C(k), and a topological version of the Rokhlin lemma holds. Furthermore, under the same assumptions, if mdim(X,ℤk)=0, then (X,ℤk) has the small boundary property. One of the applications of this theory is related to Downarowicz’s entropy structure, a master invariant for entropy theory, which captures the emergence of entropy on different scales. Indeed, we generalize this invariant and prove the Boyle–Downarowicz symbolic extension entropy theorem in the setting of ℤk-actions. This theorem describes what entropies are achievable in symbolic extensions.