Hausdorff dimension of the singularities for invariant measures of expanding dynamical systems

1988 ◽  
pp. 47-58 ◽  
Author(s):  
P. Collet
Keyword(s):  
Dynamical Systems ◽  
Hausdorff Dimension ◽  
Invariant Measures

scholarly journals Chaotically driven sigmoidal maps

2017 ◽  
Vol 18 (02) ◽  
pp. 1850009 ◽  
Author(s):  
Gerhard Keller ◽  
Atsuya Otani
Keyword(s):  
Dynamical Systems ◽  
Hausdorff Dimension ◽  
Global Attractor ◽  
Schwarzian Derivative ◽  
Invariant Measures ◽  
Skew Product ◽  
Partial Hyperbolicity ◽  
Base Points ◽  
Hyperbolicity Assumption ◽  
Uniformly Bounded

We consider skew product dynamical systems [Formula: see text] with a (generalized) baker transformation [Formula: see text] at the base and uniformly bounded increasing [Formula: see text] fibre maps [Formula: see text] with negative Schwarzian derivative. Under a partial hyperbolicity assumption that ensures the existence of strong stable fibres for [Formula: see text], we prove that the presence of these fibres restricts considerably the possible structures of invariant measures — both topologically and measure theoretically, and that this finally allows to provide a “thermodynamic formula” for the Hausdorff dimension of set of those base points over which the dynamics are synchronized, i.e. over which the global attractor consists of just one point.

Using Generalized Cell Mapping to Approximate Invariant Measures on Compact Manifolds

1997 ◽  
Vol 07 (11) ◽  
pp. 2487-2499 ◽  
Author(s):  
Rabbijah Guder ◽  
Edwin Kreuzer
Keyword(s):  
Dynamical Systems ◽  
Invariant Measures ◽  
Nonlinear Dynamical Systems ◽  
Powerful Method ◽  
Cell Mapping ◽  
Generalized Cell Mapping ◽  
Nonlinear Dynamical ◽  
Long Term Behavior ◽  
Mapping Theory

In order to predict the long term behavior of nonlinear dynamical systems the generalized cell mapping is an efficient and powerful method for numerical analysis. For this reason it is of interest to know under what circumstances dynamical quantities of the generalized cell mapping (like persistent groups, stationary densities, …) reflect the dynamics of the system (attractors, invariant measures, …). In this article we develop such connections between the generalized cell mapping theory and the theory of nonlinear dynamical systems. We prove that the generalized cell mapping is a discretization of the Frobenius–Perron operator. By applying the results obtained for the Frobenius–Perron operator to the generalized cell mapping we outline for some classes of transformations that the stationary densities of the generalized cell mapping converges to an invariant measure of the system. Furthermore, we discuss what kind of measures and attractors can be approximated by this method.

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