Geometry of Cantor Systems
نویسنده
چکیده
A Cantor system is defined. The geometry of a certain family of Cantor systems is studied. Such a family arises in dynamical systems as hyperbolicity is created. We prove that the bridge geometry of a Cantor system in such a family is uniformly bounded and that the gap geometry is regulated by the size of the leading gap.
منابع مشابه
Asymptotic Hausdorff dimensions of Cantor sets associated with an asymptotically non-hyperbolic family
The geometry of Cantor systems associated with an asymptotically non-hyperbolic family (f )0≤ ≤ 0 was studied by Jiang (Geometry of Cantor systems. Trans. Amer. Math. Soc. 351 (1999), 1975–1987). By applying the geometry studied there, we prove that the Hausdorff dimension of the maximal invariant set of f behaves like 1−K 1/γ asymptotically, as was conjectured by Jiang (Generalized Ulam–von Ne...
متن کاملRelative Kolmogorov complexity and geometry
We use the connection of Hausdorff dimension and Kolmogorov complexity to describe a geometry on the Cantor set including concepts of angle, projections and scalar multiplication. A question related to compressibility is addressed using these geometrical ideas.
متن کاملOn the Structures of Generating Iterated Function Systems of Cantor Sets
A generating IFS of a Cantor set F is an IFS whose attractor is F . For a given Cantor set such as the middle-3rd Cantor set we consider the set of its generating IFSs. We examine the existence of a minimal generating IFS, i.e. every other generating IFS of F is an iterating of that IFS. We also study the structures of the semi-group of homogeneous generating IFSs of a Cantor set F in R under t...
متن کاملInfinite iterated function systems in cantor space and the hausdorff measure of omega-power languages
We use means of formal language theory to estimate the Hausdorff measure of sets of a certain shape in Cantor space. These sets are closely related to infinite iterated function systems in fractal geometry. Our results are used to provide a series of simple examples for the noncoincidence of limit sets and attractors for infinite iterated function systems. ∗A preliminary version appeared as On ...
متن کاملA Non-additive Thermodynamic Formalism and Applications to Dimension Theory of Hyperbolic Dynamical Systems
A 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 are coded by an arbitrary symbolic dynamics, and the geometry of the construction may depend on all the s...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 1999