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thermodynamic equilibrium. A measure-preserving dynamical system is defined as a probability space and a measure-preserving transformation on it. In more detail...

mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective...

is said to be a measure-preserving transformation of X , if it is a map from X to itself, it is Σ-measurable, and is measure-preserving. The triplet (T...

they preserve the measure that is density. The formal definition is as follows: Let T : X → X be a measure-preserving transformation on a measure space...

the outcomes are independent, the shift preserves the measure, and thus T is a measure-preserving transformation. The quadruplet ( X , A , μ , T ) {\displaystyle...

of several formal mathematical definitions. For example, a measure-preserving transformation T is said to be strong mixing if lim k → ∞ μ ( T − k A ∩ B...

X {\displaystyle T:X\to X} is a (possibly noninvertible) measure-preserving transformation, and that f ∈ L 1 ( μ , R ) {\displaystyle f\in L^{1}(\mu...

\Sigma ,\mu )} be a finite measure space and let f : X → X {\displaystyle f\colon X\to X} be a measure-preserving transformation. Below are two alternative...

space, only its distribution. Such a system is "measure-preserving" (area-preserving, volume-preserving). A formal difficulty arises when one tries to...

velocity while preserving its magnitude, and exchanging primed and unprimed variables, always applies to finding the inverse transformation of every boost...

function being preserved across a monotonic transform (see also monotone preferences). In this context, the term "monotonic transformation" refers to a...

orientation-preserving orthogonal transformation). Indeed, when an orthogonal transformation matrix produces a reflection, its determinant is −1. A measure of...

\textstyle {\bigcap _{n=1}^{\infty }G_{n}}} . An invertible measure-preserving transformation on a standard probability space that obeys the 0-1 law is...

Krieger, Wolfgang (1970). "On entropy and generators of measure-preserving transformations". Transactions of the American Mathematical Society. 149 (2):...

mathematics, an invariant measure is a measure that is preserved by some function. The function may be a geometric transformation. For examples, circular...

and Wintner (1941). Suppose that τ is a measure-preserving transformation of a measure space S with finite measure. If f is a real-valued integrable function...

applies. An important special case of conservative systems are the measure-preserving dynamical systems. Informally, dynamical systems describe the time...

measurements are in metres or seconds, so is the measure of dispersion. Examples of dispersion measures include: Standard deviation Interquartile range...

any measure-preserving transformation T {\displaystyle T} of the probability space (Ω, β, μ) and A ∈ β {\displaystyle A\in \beta } of positive measure there...

Grigorchuk's group G was constructed as a group of Lebesgue-measure-preserving transformations on the unit interval, but subsequently simpler descriptions...