Hyperbolic sets, transversal homoclinic trajectories, and symbolic dynamics for C1-maps in banach spaces

1990 ◽  
Vol 2 (3) ◽  
pp. 325-365 ◽  
Author(s):  
Heinrich Steinlein ◽  
Hans-Otto Walther
Keyword(s):  
Banach Spaces ◽  
Symbolic Dynamics ◽  
Hyperbolic Sets

A non-additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems

1996 ◽  
Vol 16 (5) ◽  
pp. 871-927 ◽  
Author(s):  
Luis M. Barreira
Keyword(s):  
Symbolic Dynamics ◽  
Broad Class ◽  
Thermodynamic Formalism ◽  
Upper Bounds ◽  
Dimension Theory ◽  
Lower And Upper Bounds ◽  
Closed Sets ◽  
Hyperbolic Dynamical Systems ◽  
Hyperbolic Sets ◽  
Additive Version

AbstractA non-additive version of the thermodynamic formalism is developed. This allows us to obtain lower and upper bounds for the dimension of a broad class of Cantor-like sets. These are constructed with a decreasing sequence of closed sets that may satisfy no asymptotic behavior. Moreover, they can be coded by arbitrary symbolic dynamics, and the geometry of the construction may depend on all the symbolic past. Applications include estimates of dimension for hyperbolic sets of maps that need not be differentiable.

Fredholm Theory in Banach Spaces

1986 ◽  
Author(s):  
Anthony Francis Ruston
Keyword(s):  
Banach Spaces ◽  
Fredholm Theory

D-test: A New Test for Analyzing Scale Invariance Using Symbolic Dynamics and Symbolic Entropy

Methodology ◽  
2011 ◽  
Vol 7 (3) ◽  
pp. 88-95 ◽  
Author(s):  
Jose A. Martínez ◽  
Manuel Ruiz Marín
Keyword(s):  
Symbolic Dynamics ◽  
Scale Invariance ◽  
Rating Scales ◽  
Rating Scale ◽  
Marketing Research ◽  
Response Patterns ◽  
Measurement Scales ◽  
Improve Measurement ◽  
Entropy Measures ◽  
The Difference

The aim of this study is to improve measurement in marketing research by constructing a new, simple, nonparametric, consistent, and powerful test to study scale invariance. The test is called D-test. D-test is constructed using symbolic dynamics and symbolic entropy as a measure of the difference between the response patterns which comes from two measurement scales. We also give a standard asymptotic distribution of our statistic. Given that the test is based on entropy measures, it avoids smoothed nonparametric estimation. We applied D-test to a real marketing research to study if scale invariance holds when measuring service quality in a sports service. We considered a free-scale as a reference scale and then we compared it with three widely used rating scales: Likert-type scale from 1 to 5 and from 1 to 7, and semantic-differential scale from −3 to +3. Scale invariance holds for the two latter scales. This test overcomes the shortcomings of other procedures for analyzing scale invariance; and it provides researchers a tool to decide the appropriate rating scale to study specific marketing problems, and how the results of prior studies can be questioned.

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