Examples of $\mathcal{C}^r$ interval map with large symbolic extension entropy

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.

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Symbolic dynamics for non-uniformly hyperbolic systems

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2020.80 ◽

2020 ◽

pp. 1-68

Author(s):

YURI LIMA

Keyword(s):

Symbolic Dynamics ◽

Hyperbolic Systems ◽

Markov Partition ◽

Equilibrium Measures ◽

Markov Partitions ◽

Recent Advances ◽

Closed Orbits ◽

Symbolic Extension ◽

Uniformly Hyperbolic Systems

Abstract 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.

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Constant slope maps on the extended real line

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2017.3 ◽

2017 ◽

Vol 38 (8) ◽

pp. 3145-3169 ◽

Cited By ~ 1

Author(s):

MICHAŁ MISIUREWICZ ◽

SAMUEL ROTH

Keyword(s):

Real Line ◽

Half Line ◽

Piecewise Monotone ◽

Bounded Interval ◽

Interval Map ◽

Piecewise Continuous ◽

Constant Slope

For a transitive countably piecewise monotone Markov interval map we consider the question of whether there exists a conjugate map of constant slope. The answer varies depending on whether the map is continuous or only piecewise continuous, whether it is mixing or not, what slope we consider and whether the conjugate map is defined on a bounded interval, half-line or the whole real line (with the infinities included).

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ON THE RUELLE ZETA FUNCTION OF AN EXPANDING INTERVAL MAP

Difference Equations, Special Functions and Orthogonal Polynomials ◽

10.1142/9789812770752_0002 ◽

2007 ◽

Author(s):

JOÃO FERREIRA ALVES ◽

J. L. FACHADA

Keyword(s):

Zeta Function ◽

Interval Map ◽

Ruelle Zeta Function

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The Power Ratio and the Interval Map: Spiking Models and Extracellular Recordings

Journal of Neuroscience ◽

10.1523/jneurosci.18-23-10090.1998 ◽

1998 ◽

Vol 18 (23) ◽

pp. 10090-10104 ◽

Cited By ~ 65

Author(s):

Daniel S. Reich ◽

Jonathan D. Victor ◽

Bruce W. Knight

Keyword(s):

Power Ratio ◽

Extracellular Recordings ◽

Interval Map

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CHAOTIC BEHAVIOR OF INTERVAL MAPS AND TOTAL VARIATIONS OF ITERATES

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127404010540 ◽

2004 ◽

Vol 14 (07) ◽

pp. 2161-2186 ◽

Cited By ~ 26

Author(s):

GOONG CHEN ◽

TINGWEN HUANG ◽

YU HUANG

Keyword(s):

Nonlinear Systems ◽

Initial Data ◽

Topological Entropy ◽

Chaotic Behavior ◽

Homoclinic Orbits ◽

Oscillatory Behavior ◽

Exponential Rate ◽

Interval Maps ◽

Interval Map ◽

Sensitive Dependence

Interval maps reveal precious information about the chaotic behavior of general nonlinear systems. If an interval map f:I→I is chaotic, then its iterates fnwill display heightened oscillatory behavior or profiles as n→∞. This manifestation is quite intuitive and is, here in this paper, studied analytically in terms of the total variations of fnon subintervals. There are four distinctive cases of the growth of total variations of fnas n→∞:(i) the total variations of fnon I remain bounded;(ii) they grow unbounded, but not exponentially with respect to n;(iii) they grow with an exponential rate with respect to n;(iv) they grow unbounded on every subinterval of I.We study in detail these four cases in relations to the well-known notions such as sensitive dependence on initial data, topological entropy, homoclinic orbits, nonwandering sets, etc. This paper is divided into three parts. There are eight main theorems, which show that when the oscillatory profiles of the graphs of fnare more extreme, the more complex is the behavior of the system.

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