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# Irrational rotation

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 Title: Irrational rotation Author: World Heritage Encyclopedia Language: English Subject: Collection: Dynamical Systems Publisher: World Heritage Encyclopedia Publication Date:

### Irrational rotation

In the mathematical theory of dynamical systems, an irrational rotation is a map

T_\theta : [0,1] \rightarrow [0,1],\quad T_\theta(x) \triangleq x + \theta \mod 1

where θ is an irrational number. Under the identification of a circle with R/Z, or with the interval [0, 1] with the boundary points glued together, this map becomes a rotation of a circle by a proportion θ of a full revolution (i.e., an angle of 2πθ radians). Since θ is irrational, the rotation has infinite order in the circle group and the map Tθ has no periodic orbits.

Alternatively, we can use multiplicative notation for an irrational rotation by introducing the map

The relationship between the additive and multiplicative notations is the group isomorphism

\phi:([0,1],+) \to (S^1, \cdot) \quad \phi(x)=xe^{2\pi i\theta}.

It can be shown that φ is an isometry.

There is a strong distinction in circle rotations that depends on whether θ is rational or irrational. Rational rotations are less interesting examples of dynamical systems because if \theta = \frac{a}{b} and \gcd(a,b) = 1, then T_\theta^b(x) = x when x \isin [0,1]. It can also be shown that T_\theta^i(x) \ne x when 1 \le i < b.

## Contents

• Significance 1
• Properties 2
• Generalizations 3
• Applications 4
• References 6

## Significance

Irrational rotations form a fundamental example in the theory of dynamical systems. According to the Denjoy theorem, every orientation-preserving C2-diffeomorphism of the circle with an irrational rotation number θ is topologically conjugate to Tθ. An irrational rotation is a measure-preserving ergodic transformation, but it is not mixing. The Poincaré map for the dynamical system associated with the Kronecker foliation on a torus with angle θ is the irrational rotation by θ. C*-algebras associated with irrational rotations, known as irrational rotation algebras, have been extensively studied.

## Properties

• If θ is irrational, then the orbit of element of [0,1] under the rotation Tθ is dense in [0,1]. Therefore, irrational rotations are topologically transitive.
• If θ is irrational, then Tθ is uniquely ergodic.
• Irrational (and rational) rotations are not topologically mixing.
• Irrational rotations are ergodic with respect to the Lebesgue measure.
• Irrational rotations are uniquely ergodic, with the Lebesgue measure serving as the unique invariant probability measure.
• Suppose [a,b] ⊂ [0,1]. Since Tθ is ergodic,
\text{lim} _ {N \to \infty} \frac{1}{N} \sum_{n=0}^{N-1} \chi_{[a,b)}(T_\theta ^n (t))=b-a .

## Generalizations

• Circle rotations are examples of group translations.
• For a general orientation preserving homomorphism f of S1 to itself we call a homeomorphism F:\mathbb{R}\to \mathbb{R} a lift of f if \pi \circ F=f \circ \pi where \pi (t)=t \bmod 1 .[1]

## Applications

• Skew Products over Rotations of the Circle: In 1969[2] William A. Veech constructed examples of minimal and not uniquely ergodic dynamical systems as follows: "Take two copies of the unit circle and mark off segment J of length 2πα in the counterclockwise direction on each one with endpoint at 0. Now take θ irrational and consider the following dynamical system. Start with a point p, say in the first circle. Rotate counterclockwise by 2πθ until the first time the orbit lands in J; then switch to the corresponding point in the second circle, rotate by 2πθ until the first time the point lands in J; switch back to the first circle and so forth. Veech showed that if θ is irrational, then there exists irrational α for which this system is minimal and the Lebesgue measure is not uniquely ergodic."[3]

## References

-- Module:Hatnote -- -- -- -- This module produces hatnote links and links to related articles. It -- -- implements the and meta-templates and includes -- -- helper functions for other Lua hatnote modules. --

local libraryUtil = require('libraryUtil') local checkType = libraryUtil.checkType local mArguments -- lazily initialise Module:Arguments local yesno -- lazily initialise Module:Yesno

local p = {}

-- Helper functions

local function getArgs(frame) -- Fetches the arguments from the parent frame. Whitespace is trimmed and -- blanks are removed. mArguments = require('Module:Arguments') return mArguments.getArgs(frame, {parentOnly = true}) end

local function removeInitialColon(s) -- Removes the initial colon from a string, if present. return s:match('^:?(.*)') end

function p.findNamespaceId(link, removeColon) -- Finds the namespace id (namespace number) of a link or a pagename. This -- function will not work if the link is enclosed in double brackets. Colons -- are trimmed from the start of the link by default. To skip colon -- trimming, set the removeColon parameter to true. checkType('findNamespaceId', 1, link, 'string') checkType('findNamespaceId', 2, removeColon, 'boolean', true) if removeColon ~= false then link = removeInitialColon(link) end local namespace = link:match('^(.-):') if namespace then local nsTable = mw.site.namespaces[namespace] if nsTable then return nsTable.id end end return 0 end

function p.formatPages(...) -- Formats a list of pages using formatLink and returns it as an array. Nil -- values are not allowed. local pages = {...} local ret = {} for i, page in ipairs(pages) do ret[i] = p._formatLink(page) end return ret end

function p.formatPageTables(...) -- Takes a list of page/display tables and returns it as a list of -- formatted links. Nil values are not allowed. local pages = {...} local links = {} for i, t in ipairs(pages) do checkType('formatPageTables', i, t, 'table') local link = t[1] local display = t[2] links[i] = p._formatLink(link, display) end return links end

function p.makeWikitextError(msg, helpLink, addTrackingCategory) -- Formats an error message to be returned to wikitext. If -- addTrackingCategory is not false after being returned from -- Module:Yesno, and if we are not on a talk page, a tracking category -- is added. checkType('makeWikitextError', 1, msg, 'string') checkType('makeWikitextError', 2, helpLink, 'string', true) yesno = require('Module:Yesno') local title = mw.title.getCurrentTitle() -- Make the help link text. local helpText if helpLink then helpText = ' (help)' else helpText = end -- Make the category text. local category if not title.isTalkPage and yesno(addTrackingCategory) ~= false then category = 'Hatnote templates with errors' category = string.format( '%s:%s', mw.site.namespaces[14].name, category ) else category = end return string.format( '%s', msg, helpText, category ) end

-- Format link -- -- Makes a wikilink from the given link and display values. Links are escaped -- with colons if necessary, and links to sections are detected and displayed -- with " § " as a separator rather than the standard MediaWiki "#". Used in -- the template.

function p._formatLink(link, display) -- Find whether we need to use the colon trick or not. We need to use the -- colon trick for categories and files, as otherwise category links -- categorise the page and file links display the file. checkType('_formatLink', 1, link, 'string') checkType('_formatLink', 2, display, 'string', true) link = removeInitialColon(link) local namespace = p.findNamespaceId(link, false) local colon if namespace == 6 or namespace == 14 then colon = ':' else colon = end -- Find whether a faux display value has been added with the | magic -- word. if not display then local prePipe, postPipe = link:match('^(.-)|(.*)$') link = prePipe or link display = postPipe end -- Find the display value. if not display then local page, section = link:match('^(.-)#(.*)$') if page then display = page .. ' § ' .. section end end -- Assemble the link. if display then return string.format('%s', colon, link, display) else return string.format('%s%s', colon, link) end end

-- Hatnote -- -- Produces standard hatnote text. Implements the template.

function p.hatnote(frame) local args = getArgs(frame) local s = args[1] local options = {} if not s then return p.makeWikitextError( 'no text specified', 'Template:Hatnote#Errors', args.category ) end options.extraclasses = args.extraclasses options.selfref = args.selfref return p._hatnote(s, options) end

function p._hatnote(s, options) checkType('_hatnote', 1, s, 'string') checkType('_hatnote', 2, options, 'table', true) local classes = {'hatnote'} local extraclasses = options.extraclasses local selfref = options.selfref if type(extraclasses) == 'string' then classes[#classes + 1] = extraclasses end if selfref then classes[#classes + 1] = 'selfref' end return string.format( '
%s
', table.concat(classes, ' '), s )

end

return p-------------------------------------------------------------------------------- -- Module:Hatnote -- -- -- -- This module produces hatnote links and links to related articles. It -- -- implements the and meta-templates and includes -- -- helper functions for other Lua hatnote modules. --

local libraryUtil = require('libraryUtil') local checkType = libraryUtil.checkType local mArguments -- lazily initialise Module:Arguments local yesno -- lazily initialise Module:Yesno

local p = {}

-- Helper functions

local function getArgs(frame) -- Fetches the arguments from the parent frame. Whitespace is trimmed and -- blanks are removed. mArguments = require('Module:Arguments') return mArguments.getArgs(frame, {parentOnly = true}) end

local function removeInitialColon(s) -- Removes the initial colon from a string, if present. return s:match('^:?(.*)') end

function p.findNamespaceId(link, removeColon) -- Finds the namespace id (namespace number) of a link or a pagename. This -- function will not work if the link is enclosed in double brackets. Colons -- are trimmed from the start of the link by default. To skip colon -- trimming, set the removeColon parameter to true. checkType('findNamespaceId', 1, link, 'string') checkType('findNamespaceId', 2, removeColon, 'boolean', true) if removeColon ~= false then link = removeInitialColon(link) end local namespace = link:match('^(.-):') if namespace then local nsTable = mw.site.namespaces[namespace] if nsTable then return nsTable.id end end return 0 end

function p.formatPages(...) -- Formats a list of pages using formatLink and returns it as an array. Nil -- values are not allowed. local pages = {...} local ret = {} for i, page in ipairs(pages) do ret[i] = p._formatLink(page) end return ret end

function p.formatPageTables(...) -- Takes a list of page/display tables and returns it as a list of -- formatted links. Nil values are not allowed. local pages = {...} local links = {} for i, t in ipairs(pages) do checkType('formatPageTables', i, t, 'table') local link = t[1] local display = t[2] links[i] = p._formatLink(link, display) end return links end

function p.makeWikitextError(msg, helpLink, addTrackingCategory) -- Formats an error message to be returned to wikitext. If -- addTrackingCategory is not false after being returned from -- Module:Yesno, and if we are not on a talk page, a tracking category -- is added. checkType('makeWikitextError', 1, msg, 'string') checkType('makeWikitextError', 2, helpLink, 'string', true) yesno = require('Module:Yesno') local title = mw.title.getCurrentTitle() -- Make the help link text. local helpText if helpLink then helpText = ' (help)' else helpText = end -- Make the category text. local category if not title.isTalkPage and yesno(addTrackingCategory) ~= false then category = 'Hatnote templates with errors' category = string.format( '%s:%s', mw.site.namespaces[14].name, category ) else category = end return string.format( '%s', msg, helpText, category ) end

-- Format link -- -- Makes a wikilink from the given link and display values. Links are escaped -- with colons if necessary, and links to sections are detected and displayed -- with " § " as a separator rather than the standard MediaWiki "#". Used in -- the template.

function p._formatLink(link, display) -- Find whether we need to use the colon trick or not. We need to use the -- colon trick for categories and files, as otherwise category links -- categorise the page and file links display the file. checkType('_formatLink', 1, link, 'string') checkType('_formatLink', 2, display, 'string', true) link = removeInitialColon(link) local namespace = p.findNamespaceId(link, false) local colon if namespace == 6 or namespace == 14 then colon = ':' else colon = end -- Find whether a faux display value has been added with the | magic -- word. if not display then local prePipe, postPipe = link:match('^(.-)|(.*)$') link = prePipe or link display = postPipe end -- Find the display value. if not display then local page, section = link:match('^(.-)#(.*)$') if page then display = page .. ' § ' .. section end end -- Assemble the link. if display then return string.format('%s', colon, link, display) else return string.format('%s%s', colon, link) end end

-- Hatnote -- -- Produces standard hatnote text. Implements the template.

function p.hatnote(frame) local args = getArgs(frame) local s = args[1] local options = {} if not s then return p.makeWikitextError( 'no text specified', 'Template:Hatnote#Errors', args.category ) end options.extraclasses = args.extraclasses options.selfref = args.selfref return p._hatnote(s, options) end

function p._hatnote(s, options) checkType('_hatnote', 1, s, 'string') checkType('_hatnote', 2, options, 'table', true) local classes = {'hatnote'} local extraclasses = options.extraclasses local selfref = options.selfref if type(extraclasses) == 'string' then classes[#classes + 1] = extraclasses end if selfref then classes[#classes + 1] = 'selfref' end return string.format( '
%s
', table.concat(classes, ' '), s )

end

return p
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