A Non-additive Thermodynamic Formalism and Applications to Dimension Theory of Hyperbolic Dynamical Systems
نویسنده
چکیده
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 symbolic past. Applications include estimates of dimension for hyperbolic sets of maps that need not be differentiable.
منابع مشابه
A Non-additive Thermodynamic Formalism and Dimension Theory of Hyperbolic Dynamical Systems
Introduction We begin with the description of a geometric construction in the real line. We consider p positive numbers λ1, . . . , λp < 1, and choose p disjoint closed intervals ∆1, . . . , ∆p with length λ1, . . . , λp. For each k = 1, . . . , p, we choose again p disjoint closed intervals ∆k1, . . . , ∆kp ⊂ ∆k with length λkλ1, . . . , λkλp. Iterating this procedure, for each integer n we ob...
متن کاملHyperbolicity and Recurrence in Dynamical Systems: a Survey of Recent Results
We discuss selected topics of current research interest in the theory of dynamical systems, with emphasis on dimension theory, multifractal analysis, and quantitative recurrence. The topics include the quantitative versus the qualitative behavior of Poincaré recurrence, the product structure of invariant measures and return times, the dimension of invariant sets and invariant measures, the comp...
متن کاملHistoric set carries full hausdorff dimension
We prove that the historic set for ratio of Birkhoff average is either empty or full of Hausdorff dimension in a class of one dimensional non-uniformly hyperbolic dynamical systems.
متن کاملDynamical Behavior of a Rigid Body with One Fixed Point (Gyroscope). Basic Concepts and Results. Open Problems: a Review
The study of the dynamic behavior of a rigid body with one fixed point (gyroscope) has a long history. A number of famous mathematicians and mechanical engineers have devoted enormous time and effort to clarify the role of dynamic effects on its movement (behavior) – stable, periodic, quasi-periodic or chaotic. The main objectives of this review are: 1) to outline the characteristic features of...
متن کامل