Dynamical systems in a Euclidean space

Geometry of orbit is a subject of many investigations because it has important role in many branches of mathematics such as dynamical systems, control theory. In this paper it is studied geometry of orbits of conformal vector fields. It is shown that orbits of conformal vector fields are integral submanifolds of completely integrable distributions. Also for Euclidean space it is proven that if all orbits have the same dimension they are closed subsets.

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A COMPUTATIONAL ERGODIC THEOREM FOR INFINITE ITERATED FUNCTION SYSTEMS

Stochastics and Dynamics ◽

10.1142/s0219493708002354 ◽

2008 ◽

Vol 08 (03) ◽

pp. 365-381 ◽

Cited By ~ 3

Author(s):

NGUYEN DINH CONG ◽

DOAN THAI SON ◽

STEFAN SIEGMUND

Keyword(s):

Dynamical Systems ◽

Euclidean Space ◽

Ergodic Theorem ◽

Error Bound ◽

Metric Space ◽

Iterated Function Systems ◽

Random Dynamical Systems ◽

Compact Metric Space ◽

Iterated Function ◽

Time Average

Iterated function systems are examples of random dynamical systems and became popular as generators of fractals like the Sierpinski Gasket and the Barnsley Fern. In this paper we prove an ergodic theorem for iterated function systems which consist of countably many functions and which are contractive on average on an arbitrary compact metric space and we provide a computational version of this ergodic theorem in Euclidean space which allows to numerically approximate the time average together with an explicit error bound. The results are applied to an explicit example.

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On a class of topological conjugacy with a homothety

Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva ◽

10.15507/2079-6900.22.202003.261-267 ◽

2020 ◽

Vol 22 (3) ◽

pp. 261-267

Author(s):

Elena Ya. Gurevich ◽

Aleksey A. Makarov

Keyword(s):

Dynamical Systems ◽

Euclidean Space ◽

20Th Century ◽

Classical Theory ◽

Topological Conjugacy ◽

Smooth Case

We consider a class H(Rn) of orientation-preserving homeomorphisms of Euclidean space Rn such that for any homeomorphism h∈H(Rn) and for any point x∈Rn a condition limn→+∞hn(x)→O holds, were O is the origin. It is proved that for any n≥1 an arbitrary homeomorphism h∈H(Rn) is topologically conjugated with the homothety an:Rn→Rn, given by an(x1,…,an)=(12x1,…,12xn). For a smooth case under the condition that all eigenvalues of the differential of the mapping h have absolute values smaller than one, this fact follows from the classical theory of dynamical systems. In the topological case for n∉{4,5} this fact is proven in several works of 20th century, but authors do not know any papers where it would be proven for n∈{4,5}. This paper fills this gap.

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Ergodic fractal measures and dimension conservation

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385708000084 ◽

2008 ◽

Vol 28 (2) ◽

pp. 405-422 ◽

Cited By ~ 41

Author(s):

HILLEL FURSTENBERG

Keyword(s):

Dynamical Systems ◽

Hausdorff Dimension ◽

Euclidean Space ◽

Ergodic Theorem ◽

Compact Set ◽

Linear Map ◽

Compactly Supported ◽

Almost Everywhere ◽

Fractal Measures

AbstractA linear map from one Euclidean space to another may map a compact set bijectively to a set of smaller Hausdorff dimension. For ‘homogeneous’ fractals (to be defined), there is a phenomenon of ‘dimension conservation’. In proving this we shall introduce dynamical systems whose states represent compactly supported measures in which progression in time corresponds to progressively increasing magnification. Application of the ergodic theorem will show that, generically, dimension conservation is valid. This ‘almost everywhere’ result implies a non-probabilistic statement for homogeneous fractals.

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