''of a dynamical system''
[[Category:Ergodic theory]]
1 KB (242 words) - 21:18, 8 November 2014
$#C+1 = 14 : ~/encyclopedia/old_files/data/H047/H.0407820 Homology of a dynamical system,
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4 KB (528 words) - 22:10, 5 June 2020
A [[flow (continuous-time dynamical system)]] the trajectories of which become almost-periodic after a certain
In the above, an almost-periodic trajectory in a [[dynamical system]] $\{ S^t \}$ on a metric space $(X,\rho)$ is the trajectory of any
1,020 bytes (154 words) - 19:16, 25 October 2014
$#C+1 = 7 : ~/encyclopedia/old_files/data/M062/M.0602950 Maximal ergodic theorem
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2 KB (340 words) - 08:12, 9 January 2024
...our of the system resembles that of a [[Gradient dynamical system|gradient dynamical system]] (see [[#References|[1]]], [[#References|[2]]]).
...valign="top"> A.N. Sharkovskii, V.A. Dobrynskii, , ''Dynamical systems and problems of stability of solutions of differential equations'' , Kiev (197
3 KB (395 words) - 10:06, 24 August 2014
''$\{T_t\}$ with phase space $X$ and invariant measure $\mu$
A common name for various spectral invariants and spectral properties of the corresponding group (or semi-group) of unitary (
5 KB (710 words) - 12:58, 16 July 2014
[[Category:Ergodic theorems, spectral theory, Markov operators]]
...$ T $ of a $ \sigma $-finite measure space $ (X,\Sigma,\mu) $, Birkhoff’s ergodic theorem states that for any function $ f \in {L^{1}}(X,\Sigma,\mu) $, the l
3 KB (460 words) - 05:51, 29 November 2016
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...re|measure]] when $X$ is a shift space (cf. [[Shift dynamical system|Shift dynamical system]]). The importance of absolutely continuous invariant measures is du
5 KB (722 words) - 16:46, 1 July 2020
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[[Category:Measure-theoretic ergodic theory]]
2 KB (260 words) - 08:00, 6 June 2020
$#C+1 = 22 : ~/encyclopedia/old_files/data/O068/O.0608360 Operator ergodic theorem
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5 KB (685 words) - 16:32, 6 January 2024
$#C+1 = 36 : ~/encyclopedia/old_files/data/V096/V.0906910 von Neumann ergodic theorem
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4 KB (632 words) - 17:38, 6 January 2024
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[[Category:Ergodic theory]]
5 KB (696 words) - 08:01, 6 June 2020
$#C+1 = 34 : ~/encyclopedia/old_files/data/S084/S.0804890 Shift dynamical system
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6 KB (862 words) - 08:13, 6 June 2020
...31 : ~/encyclopedia/old_files/data/F040/F.0400650 Flow (continuous\AAhtime dynamical system)
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3 KB (541 words) - 19:39, 5 June 2020
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...nd [[Topological dynamics|topological dynamics]] used in phrases like: "a dynamical system (flow, cascade or transformation generating the latter) has a quasi-
7 KB (978 words) - 05:56, 19 March 2022
...t\}$ (of a [[Flow (continuous-time dynamical system)|flow (continuous-time dynamical system)]] or of a [[Cascade|cascade]]) with phase space $X$''
...trarily large countable transfinite number even for flows in $\mathbf R^3$ and on certain open surfaces (see [[#References|[8]]]–[[#References|[10]]]).
4 KB (617 words) - 16:07, 19 August 2014
$#C+1 = 71 : ~/encyclopedia/old_files/data/E035/E.0305760 Entropy theory of a dynamical system
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7 KB (1,040 words) - 19:23, 13 January 2024
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is the partition into points; and c) the only measurable partition $ \mathop{\rm mod} 0 $
7 KB (1,037 words) - 16:58, 1 July 2020
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...eir generalization — a sequence of independent trials with the same result and with the same probability distribution.
4 KB (566 words) - 10:58, 29 May 2020
...somorphic, and therefore one often does not distinguish between the states and the phase points that represent them.
...ary to consider statistical states of systems; for classical (non-quantum) systems they are described by probability distributions on the phase space. The rul
6 KB (906 words) - 21:49, 30 April 2014
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''of a dynamical system''
11 KB (1,535 words) - 17:57, 29 October 2020
...$M$, the difference between its "geometric" trajectory in ordinary space and the phase trajectory is still retained, since the state is not just the geo
...that is not necessarily a curve) corresponds to the abstract concept of a dynamical system as a group or semi-group of transformations, not necessarily a one-p
2 KB (357 words) - 16:50, 15 September 2014
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''Wiener–Wintner ergodic theorem''
9 KB (1,431 words) - 17:03, 1 July 2020
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''of a dynamical system''
7 KB (975 words) - 07:09, 22 March 2023
''metric theory of dynamical systems''
[[Category:Ergodic theory]]
13 KB (2,012 words) - 22:03, 6 April 2012
...nal space. The [[Lebesgue measure]] of a set does not change as it evolves and can be identified with its [[Probability|probability]].
...ract the statistical properties (''e.g.'', ignoring sets of probability 0) and regard the state-space as an abstract [[Measure space|measure space]]. Equi
8 KB (1,391 words) - 13:02, 12 December 2013
...orem that permits one to establish the absence of closed trajectories of [[dynamical system]]s in the plane, defined by the equation
...ected domain in the $(x,y)$-plane, if the functions $P$ and $Q\in C^1(G)$, and if a function $f(x,y)\in C^1(G)$ can be found such that
2 KB (364 words) - 15:57, 14 February 2020
...n is the case where the [[Phase space|phase space]] is a metric compactum, and time runs through $\mathbf R$, $\mathbf Z$ or $\mathbf N$ (this is assumed
...stability]], non-wanderingness (cf. [[Wandering point|Wandering point]]), and [[Chain recurrence|chain recurrence]]. As time increases, any trajectory ap
12 KB (1,685 words) - 22:03, 10 December 2017
...D.V. Anosov (see , [[#References|[2]]]), and they are often called Anosov systems.
...ime averages to the space average is subject to the central limit theorem, and the rate of intermixing is exponential ("exponential decay of correlations"
9 KB (1,321 words) - 07:59, 21 June 2014
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'' $ \{ f ^ { t } x \} $ of a dynamical system $ f ^ { t } $''
5 KB (812 words) - 04:31, 14 September 2022
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is a bijection and the mapping $ T ^ {-} 1 $
10 KB (1,584 words) - 22:13, 5 June 2020
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A property defined for a [[Topological dynamical system|topological dynamical system]] $ \{ T _ {t} \} $,
3 KB (470 words) - 08:25, 6 June 2020
''of a dynamical system $\{T_t\}$ with a (quasi-) invariant measure $\mu$''
[[Category:Ergodic theory]]
9 KB (1,352 words) - 10:41, 9 November 2014
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...c theory]] analogous to the [[Metric entropy|metric entropy]] of dynamical systems (introduced in [[#References|[1]]]). For an open covering $ \mathfrak A $
10 KB (1,429 words) - 16:40, 29 October 2023
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...tems (or models describing deterministic evolution, cf. [[Dynamical system|Dynamical system]]) considered are differential equations $ \dot{x} = F ( x) $,
8 KB (1,192 words) - 16:43, 4 June 2020
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One of the methods of [[Ergodic theory|ergodic theory]]. Any automorphism $ T $
8 KB (1,151 words) - 12:12, 21 March 2022
...le of an invariant subset of the topological Bernoulli automorphism. The [[dynamical system]] in $\Omega_1$ generated by the shift $\sigma{\downharpoonright}_{\
...systems which themselves are defined completely independently of $\Omega$ and $\sigma$.
15 KB (2,197 words) - 08:48, 29 April 2023
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A class of dynamical systems (cf. [[Dynamical system|Dynamical system]]). An example is the flow generated by all translations of a torus
12 KB (1,841 words) - 18:33, 5 June 2020
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and $ \beta $
3 KB (367 words) - 17:33, 5 June 2020
$#C+1 = 481 : ~/encyclopedia/old_files/data/P110/P.1100120 Pesin theory
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39 KB (5,692 words) - 14:54, 7 June 2020
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''software for dynamical systems''
13 KB (1,928 words) - 17:00, 1 July 2020
...property is called almost-periodicity of the point $w$. If $F=W$, then the dynamical system itself is called minimal.
...e said that minimality of a dynamical system places no restrictions on its ergodic properties with respect to any of its invariant measures, [[#References|[7]
10 KB (1,453 words) - 17:20, 27 October 2014
$#C+1 = 121 : ~/encyclopedia/old_files/data/D034/D.0304280 Dynamical system
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27 KB (4,058 words) - 19:36, 5 June 2020
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...[[Invariant measure|invariant measure]] (cf. also [[Ergodic theory|Ergodic theory]]).
9 KB (1,389 words) - 17:52, 6 January 2024
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and assuming values $ x _ {1} , x _ {2} \dots $
7 KB (1,067 words) - 13:06, 6 January 2024
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unknowns $ p = ( p _ {1} \dots p _ {n} ) $ ("generalized momenta") and $ q = ( q _ {1} \dots q _ {n} ) $ ("generalized coordinates"), of the for
13 KB (1,837 words) - 07:56, 21 January 2024
A sequence appearing in numerous fields and discovered and rediscovered in different contexts.
...erated by an iterative process called substitution. Let $\calA = \{a, b\}$ and let $\calA^*$ denote the set of words defined on the
10 KB (1,633 words) - 08:05, 26 November 2023
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...ntioned below and is asymptotically stable (that is, it is Lyapunov stable and all trajectories in some neighbourhood of $ L $
10 KB (1,485 words) - 04:11, 6 June 2020
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and $ f _ {x} ^ { \prime } $
15 KB (2,177 words) - 16:07, 5 February 2022
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and every $ x,y \in B ^ {n} $
4 KB (636 words) - 20:31, 17 January 2024