迭代函数

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In mathematics, an iterated function is a function $X \to X$ (that is, a function from some set $X$ to itself) which is obtained by composing another function $f : X \to X$ with itself a certain number of times. The process of repeatedly applying the same function is called iteration. In this process, starting from some initial number, the result of applying a given function is fed again in the function as input, and this process is repeated.

In mathematics, an iterated function is a function (that is, a function from some set to itself) which is obtained by composing another function with itself a certain number of times. The process of repeatedly applying the same function is called iteration. In this process, starting from some initial number, the result of applying a given function is fed again in the function as input, and this process is repeated.

在数学中，迭代函数是一个 $X \to X$ 的函数(也就是从一个集合映射到它本身的一个函数) ，它以类似“二人转”的方式与另一个函数组合使用，被调用函数的输出再次作为迭代函数的输入，多次循环执行。这个重复地与自身复合的过程叫做迭代。

Iterated functions are objects of study in computer science, fractals, dynamical systems, mathematics and renormalization group physics.

迭代函数是计算机科学、分形、动力系统、数学和重整化群物理学的研究对象。

Definition

The formal definition of an iterated function on a set X follows.

在集合X上的迭代函数的正式定义如下：

Let $X$ be a set and $f : X \to X$ be a function.

Let be a set and be a function.

设 $X$ 是一个集合， $f : X \to X$ 是一个函数。

Define $f n$ as the n-th iterate of $f$ , where n is a non-negative integer, by:

Define as the n-th iterate of , where n is a non-negative integer, by:

定义 $f n$ 为 $f$ 的第 n 次迭代，其中 n 是一个非负整数。有：

[math]\displaystyle{ f^0 ~ \stackrel{\mathrm{def}}{=} ~ \operatorname{id}_X }[/math]

[math]\displaystyle{ f^0 ~   \stackrel{\mathrm{def}}{=}  ~ \operatorname{id}_X }[/math]

[数学] f ^ 0 ~ stackrel { mathrm { def }{ = } ~ operatorname { id } x </math >

and

和

[math]\displaystyle{ f^{n+1} ~ \stackrel{\mathrm{def}}{=} ~ f \circ f^{n}, }[/math]

[math]\displaystyle{ f^{n+1} ~ \stackrel{\mathrm{def}}{=} ~ f \circ f^{n}, }[/math]

< math > f ^ { n + 1} ~ stackrel { mathrm { def }{ = } ~ f circ ^ { n } ，</math >

where $id X$ is the identity function on $X$ and $f ○ g$ denotes function composition. That is,

where is the identity function on and denotes function composition. That is,

其中 $id X$ 表示X上的恒等式， $f ○ g$ 表示复合函数。即：

(f ○ g)(x) = f (g (x))

Because the notation $f n$ may refer to both iteration (composition) of the function $f$ or exponentiation of the function $f$ (the latter is commonly used in trigonometry), some mathematicians choose to write $f ° n$ for the n-th iterate of the function $f$ .

Because the notation may refer to both iteration (composition) of the function or exponentiation of the function (the latter is commonly used in trigonometry), some mathematicians choose to write for the n-th iterate of the function .

由于 $f n$ 既可以表示函数 $f$ 的迭代(复合)，也可以表示 $f$ 的指数(后者通常用于三角函数) ，一些数学家选择 $f ° n$ 作为函数 $f$ 的第 n 次迭代的表示法。

Abelian property and iteration sequences

In general, the following identity holds for all non-negative integers $m$ and $n$ ,

In general, the following identity holds for all non-negative integers and ,

一般来说，下面的恒等式适用于所有非负整数 $m$ 和 $n$ ,

[math]\displaystyle{ f^m \circ f^n = f^n \circ f^m = f^{m+n}~. }[/math]

[math]\displaystyle{ f^m \circ f^n =   f^n \circ f^m = f^{m+n}~. }[/math]

< math > f ^ m circ ^ n = f ^ n circ f ^ m = f ^ { m + n } ~ </math >

This is structurally identical to the property of exponentiation that $a m a n = a m + n$

This is structurally identical to the property of exponentiation that a^m + n}}

这在结构上等同于 $a m a n = a m + n$

In general, for arbitrary general (negative, non-integer, etc.) indices $m$ and $n$ , this relation is called the translation functional equation, cf. Schröder's equation and Abel equation. On a logarithmic scale, this reduces to the nesting property of Chebyshev polynomials, $T m (T n (x)) = T m n (x)$ , since $T n (x) = cos(n arcos(x))$ .

In general, for arbitrary general (negative, non-integer, etc.) indices and , this relation is called the translation functional equation, cf. Schröder's equation and Abel equation. On a logarithmic scale, this reduces to the nesting property of Chebyshev polynomials, T_{m n}(x)}}, since cos(n arcos(x ))}}.

一般来说，对于任意的(负数，非整数等)指数 $m$ and $n$ ，这种关系被称为翻译函数方程。它在对数尺度上降低了切比雪夫多项式的嵌套属性。 $T m (T n (x)) = T m n (x)$ , 因此 $T n (x) = cos(n arcos(x))$ .

The relation $(f m) n (x) = (f n) m (x) = f mn (x)$ also holds, analogous to the property of exponentiation that $(a m) n = (a n) m = a mn$ .

The relation (fⁿ )^m(x) f^mn(x)}} also holds, analogous to the property of exponentiation that (aⁿ )^m a^mn}}.

关系式 $(f m) n (x) = (f n) m (x) = f mn (x)$ 同样成立，类似于公式 $(a m) n = (a n) m = a mn$ 的幂运算性质。

The sequence of functions $f n$ is called a Picard sequence,^[1]^[2] named after Charles Émile Picard.

The sequence of functions is called a Picard sequence, named after Charles Émile Picard.

这个函数序列被称为 Picard 序列，以埃米尔·皮卡命名。

For a given $x$ in $X$ , the sequence of values $f n (x)$ is called the orbit of $x$ .

For a given in , the sequence of values is called the orbit of .

对于给定集合 $X$ 的 $x$ ， $f n (x)$ 的值的序列称为 $x$ 的域。

If $f n (x) = f n + m (x)$ for some integer $m$ , the orbit is called a periodic orbit. The smallest such value of $m$ for a given $x$ is called the period of the orbit. The point $x$ itself is called a periodic point. The cycle detection problem in computer science is the algorithmic problem of finding the first periodic point in an orbit, and the period of the orbit.

If f ^n+m (x)}} for some integer , the orbit is called a periodic orbit. The smallest such value of for a given is called the period of the orbit. The point itself is called a periodic point. The cycle detection problem in computer science is the algorithmic problem of finding the first periodic point in an orbit, and the period of the orbit.

对于某个整数，如果 $f n (x) = f n + m (x)$ ，则轨道称为周期轨道。对于给定的轨道，这样的最小值叫做轨道周期，这个点本身叫做周期点。计算机科学中的循环检测问题是在轨道上找到第一个周期点以及轨道周期的计算问题。

Fixed points

If f(x) = x for some x in X (that is, the period of the orbit of x is 1), then x is called a fixed point of the iterated sequence. The set of fixed points is often denoted as Fix(f ). There exist a number of fixed-point theorems that guarantee the existence of fixed points in various situations, including the Banach fixed point theorem and the Brouwer fixed point theorem.

If f(x) = x for some x in X (that is, the period of the orbit of x is 1), then x is called a fixed point of the iterated sequence. The set of fixed points is often denoted as Fix(f ). There exist a number of fixed-point theorems that guarantee the existence of fixed points in various situations, including the Banach fixed point theorem and the Brouwer fixed point theorem.

如果对于 x 中的某些 x，f (x) = x (即 x 的轨道周期为1) ，则称 x 为迭代序列的不动点。不动点的集合通常被称为 Fix (f)。不动点定理，包括 Banach 不动点定理和 Brouwer 不动点定理。

There are several techniques for convergence acceleration of the sequences produced by fixed point iteration.^[3] For example, the Aitken method applied to an iterated fixed point is known as Steffensen's method, and produces quadratic convergence.

There are several techniques for convergence acceleration of the sequences produced by fixed point iteration. For example, the Aitken method applied to an iterated fixed point is known as Steffensen's method, and produces quadratic convergence.

对于不动点迭代产生的序列，有几种加速收敛的方法。例如，应用于迭代不动点的艾特肯方法被称为 Steffensen 方法，它可以产生二次收敛。

Limiting behaviour

Upon iteration, one may find that there are sets that shrink and converge towards a single point. In such a case, the point that is converged to is known as an attractive fixed point. Conversely, iteration may give the appearance of points diverging away from a single point; this would be the case for an unstable fixed point.^[4]

Upon iteration, one may find that there are sets that shrink and converge towards a single point. In such a case, the point that is converged to is known as an attractive fixed point. Conversely, iteration may give the appearance of points diverging away from a single point; this would be the case for an unstable fixed point.

在迭代过程中，人们可能会发现有一些集合会收缩并向一个点收敛。在这种情况下，收敛到的点被称为吸引不动点。相反，迭代可能给出点从单点发散的外观，这是不稳定不动点的情况。

When the points of the orbit converge to one or more limits, the set of accumulation points of the orbit is known as the limit set or the ω-limit set.

When the points of the orbit converge to one or more limits, the set of accumulation points of the orbit is known as the limit set or the ω-limit set.

当轨道点收敛到一个或多个极限时，轨道的累积点集称为极限集或极限集。

The ideas of attraction and repulsion generalize similarly; one may categorize iterates into stable sets and unstable sets, according to the behaviour of small neighborhoods under iteration. (Also see Infinite compositions of analytic functions.)

吸引和排斥的概念类似地进行了归纳; 根据迭代下小邻域的行为，可以将迭代分为稳定集和不稳定集。(另见解析函数的无穷组合。)

Other limiting behaviours are possible; for example, wandering points are points that move away, and never come back even close to where they started.

其他限制行为也是可能的; 例如，游荡点是指移开的点，而且永远不会回到它们开始的地方。

Invariant measure

If one considers the evolution of a density distribution, rather than that of individual point dynamics, then the limiting behavior is given by the invariant measure. It can be visualized as the behavior of a point-cloud or dust-cloud under repeated iteration. The invariant measure is an eigenstate of the Ruelle-Frobenius-Perron operator or transfer operator, corresponding to an eigenvalue of 1. Smaller eigenvalues correspond to unstable, decaying states.

如果考虑密度分布的演化，而不是单个点动态的演化，那么极限行为是由不变测度给出的。它可以可视化为点云或尘埃云在重复迭代下的行为。不变测度是 Ruelle-Frobenius-Perron 算子或转移算子的一个本征态，对应于1的本征值。较小的本征值对应不稳定的衰减状态。

In general, because repeated iteration corresponds to a shift, the transfer operator, and its adjoint, the Koopman operator can both be interpreted as shift operators action on a shift space. The theory of subshifts of finite type provides general insight into many iterated functions, especially those leading to chaos.

一般来说，由于重复迭代对应于移位、转移算子及其伴随算子，所以 Koopman 算子都可以解释为移位空间上的移位算子作用。有限型子移位理论为许多迭代函数，特别是那些导致混沌的函数提供了一般性的见解。

Fractional iterates and flows, and negative iterates

The notion $f 1/ n$ must be used with care when the equation $g n (x) = f (x)$ has multiple solutions, which is normally the case, as in Babbage's equation of the functional roots of the identity map. For example, for $n = 2$ and $f (x) = 4 x - 6$ , both $g (x) = 6 - 2 x$ and $g (x) = 2 x - 2$ are solutions; so the expression $f ½ (x)$ doesn't denote a unique function, just as numbers have multiple algebraic roots. The issue is quite similar to division by zero. The roots chosen are normally the ones belonging to the orbit under study.

The notion must be used with care when the equation f(x)}} has multiple solutions, which is normally the case, as in Babbage's equation of the functional roots of the identity map. For example, for 2}} and 4x − 6}}, both 6 − 2x}} and 2x − 2}} are solutions; so the expression doesn't denote a unique function, just as numbers have multiple algebraic roots. The issue is quite similar to division by zero. The roots chosen are normally the ones belonging to the orbit under study.

当方程 f (x)}有多个解时，这个概念必须谨慎使用，这通常是这种情况，例如巴贝奇的同一映射的函数根方程。例如，对于2}和4x-6} ，6-2x }和2x-2}都是解; 所以表达式不表示唯一的函数，就像数字有多个代数根一样。这个问题与被零除非常相似。所选择的根通常属于所研究的轨道。

Fractional iteration of a function can be defined: for instance, a half iterate of a function $f$ is a function $g$ such that $g (g (x)) = f (x)$ .^[5] This function $g (x)$ can be written using the index notation as $f ½ (x)$ . Similarly, $f ⅓ (x)$ is the function defined such that $f ⅓ (f ⅓ (f ⅓ (x))) = f (x)$ , while $f ⅔ (x)$ may be defined as equal to $f ⅓ (f ⅓ (x))$ , and so forth, all based on the principle, mentioned earlier, that $f m ○ f n = f m + n$ . This idea can be generalized so that the iteration count $n$ becomes a continuous parameter, a sort of continuous "time" of a continuous orbit.^[6]^[7]

Fractional iteration of a function can be defined: for instance, a half iterate of a function is a function such that f(x)}}. This function can be written using the index notation as . Similarly, is the function defined such that f(x)}}, while may be defined as equal to , and so forth, all based on the principle, mentioned earlier, that f^{m + n}}}. This idea can be generalized so that the iteration count becomes a continuous parameter, a sort of continuous "time" of a continuous orbit.

函数的分数次迭代可以定义为: 例如，函数的半次迭代是一个函数，使得 f (x)}}。此函数可以使用索引表示法编写，如下所示。类似地，函数是否定义为 f (x)}} ，而可能定义为等于，等等，都基于前面提到的原则，即 f m + n }。这种思想可以推广，使迭代计数成为一个连续参数，一种连续轨道的连续“时间”。

In such cases, one refers to the system as a flow. (cf. Section on conjugacy below.)

在这种情况下，人们将系统称为流。(参考《。下文关于共轭性的章节。)

Negative iterates correspond to function inverses and their compositions. For example, $f -1 (x)$ is the normal inverse of $f$ , while $f -2 (x)$ is the inverse composed with itself, i.e. $f -2 (x) = f -1 (f -1 (x))$ . Fractional negative iterates are defined analogously to fractional positive ones; for example, $f -½ (x)$ is defined such that $f - ½ (f -½ (x)) = f -1 (x)$ , or, equivalently, such that $f -½ (f ½ (x)) = f 0 (x) = x$ .

Negative iterates correspond to function inverses and their compositions. For example, is the normal inverse of , while is the inverse composed with itself, i.e. f⁻¹(f⁻¹(x))}}. Fractional negative iterates are defined analogously to fractional positive ones; for example, is defined such that f⁻¹(x)}}, or, equivalently, such that f⁰(x) x}}.

负迭代对应于函数逆和它们的组合。例如，is the normal inverse of，while is the inverse with itself，is。F & minus; 1 (f & minus; 1 (x))}.分数负迭代的定义类似于分数正迭代; 例如，定义为 f & minus; 1 (x)} ，或者等价地定义为 f 0 (x) x }。

Some formulas for fractional iteration

One of several methods of finding a series formula for fractional iteration, making use of a fixed point, is as follows.^[8]

One of several methods of finding a series formula for fractional iteration, making use of a fixed point, is as follows.

利用不动点找到分式迭代的级数公式的几种方法之一如下。

First determine a fixed point for the function such that $f (a) = a$ .

First determine a fixed point for the function such that  a}} .

首先为函数确定一个固定点，使得 a }。

Define $f n (a) = a$ for all n belonging to the reals. This, in some ways, is the most natural extra condition to place upon the fractional iterates.

Define    a}} for all n belonging to the reals. This, in some ways, is the most natural extra condition to place upon the fractional iterates.

为所有属于实数的 n 定义一个}}。在某些方面，这是放置在小数迭代上的最自然的额外条件。

Expand $f n (x)$ around the fixed point a as a Taylor series,

Expand    around the fixed point a as a Taylor series,

围绕定点 a 展开为泰勒级数,

[math]\displaystyle{ \lt math\gt 《数学》 f^n(x) = f^n(a) + (x-a)\left.\frac{d}{dx}f^n(x)\right|_{x=a} + \frac{(x-a)^2}2\left.\frac{d^2}{dx^2}f^n(x)\right|_{x=a} +\cdots f^n(x) = f^n(a) + (x-a)\left.\frac{d}{dx}f^n(x)\right|_{x=a} + \frac{(x-a)^2}2\left.\frac{d^2}{dx^2}f^n(x)\right|_{x=a} +\cdots F ^ n (x) = f ^ n (a) + (x-a) left. frac { d }{ dx } f ^ n (x) right | { x = a } + frac {(x-a) ^ 2}2 left. frac { d ^ 2}{ dx ^ 2} f ^ n (x) right | { x = a } + cdots }[/math]

</math>

数学

Expand out

Expand out

向外扩张

[math]\displaystyle{ \lt math\gt 《数学》 f^n(x) = f^n(a) + (x-a) f'(a)f'(f(a))f'(f^2(a))\cdots f'(f^{n-1}(a)) + \cdots f^n(x) = f^n(a) + (x-a) f'(a)f'(f(a))f'(f^2(a))\cdots f'(f^{n-1}(a)) + \cdots F ^ n (x) = f ^ n (a) + (x-a) f’(a) f’(f (a)) f’(f ^ 2(a))) cdots’(f ^ { n-1}(a)) + cdots }[/math]

</math>

数学

Substitute in for $f k (a)= a$ , for any k,

 Substitute in for    a}}, for any k,

代入 a } ，代表任意 k,

[math]\displaystyle{ \lt math\gt 《数学》 f^n(x) = a + (x-a) f'(a)^n + \frac{(x-a)^2}2(f''(a)f'(a)^{n-1})\left(1+f'(a)+\cdots+f'(a)^{n-1} \right)+\cdots f^n(x) = a + (x-a) f'(a)^n + \frac{(x-a)^2}2(f(a)f'(a)^{n-1})\left(1+f'(a)+\cdots+f'(a)^{n-1} \right)+\cdots F ^ n (x) = a + (x-a) f’(a) ^ n + frac {(x-a) ^ 2}2(f (a) f’(a) ^ { n-1})左(1 + f’(a) + cdots + f’(a) ^ { n-1}右) + cdots }[/math]

</math>

数学

Make use of the geometric progression to simplify terms,

Make use of the geometric progression to simplify terms,

利用等比数列来简化术语,

[math]\displaystyle{ \lt math\gt 《数学》 f^n(x) = a + (x-a) f'(a)^n + \frac{(x-a)^2}2(f''(a)f'(a)^{n-1})\frac{f'(a)^n-1}{f'(a)-1}+\cdots f^n(x) = a + (x-a) f'(a)^n + \frac{(x-a)^2}2(f(a)f'(a)^{n-1})\frac{f'(a)^n-1}{f'(a)-1}+\cdots F ^ n (x) = a + (x-a) f’(a) ^ n + frac {(x-a) ^ 2}2(f (a) f’(a) ^ { n-1}) frac { f’(a) ^ n-1}{ f’(a)-1} + cdots }[/math]

</math>

数学

There is a special case when $f'(a) = 1$ ,

There is a special case when 1}},

有一个特殊的例子:

[math]\displaystyle{ \lt math\gt 《数学》 f^n(x) = x + \frac{(x-a)^2}2(n f''(a))+ \frac{(x-a)^3}6\left(\frac{3}{2}n(n-1) f''(a)^2 + n f'''(a)\right)+\cdots f^n(x) = x + \frac{(x-a)^2}2(n f(a))+ \frac{(x-a)^3}6\left(\frac{3}{2}n(n-1) f(a)^2 + n f(a)\right)+\cdots F ^ n (x) = x + frac {(x-a) ^ 2}2(n f (a)) + frac {(x-a) ^ 3}6 left (frac {3}{2}{ n-1) f (a) ^ 2 + n f (a) right) + cdots }[/math]

</math>

数学

This can be carried on indefinitely, although inefficiently, as the latter terms become increasingly complicated. A more systematic procedure is outlined in the following section on Conjugacy.

This can be carried on indefinitely, although inefficiently, as the latter terms become increasingly complicated. A more systematic procedure is outlined in the following section on Conjugacy.

这种情况可以无限期地继续下去，尽管效率低下，因为后者的条件变得越来越复杂。下面关于共轭的章节概述了一个更加系统的程序。

Example 1

For example, setting $f (x) = Cx + D$ gives the fixed point $a = D /(1 - C)$ , so the above formula terminates to just

For example, setting Cx + D}} gives the fixed point D/(1 − C)}}, so the above formula terminates to just

例如，设置 Cx + d }给出了定点 d/(1-c)}} ，所以上面的公式结束为

[math]\displaystyle{ \lt math\gt 《数学》 f^n(x)=\frac{D}{1-C} + (x-\frac{D}{1-C})C^n=C^nx+\frac{1-C^n}{1-C}D ~, f^n(x)=\frac{D}{1-C} + (x-\frac{D}{1-C})C^n=C^nx+\frac{1-C^n}{1-C}D ~, F ^ n (x) = frac { d } + (x-frac { d }{1-C }) c ^ n = c ^ nx + frac {1-C ^ n }{1-C } d ~ , }[/math]

</math>

数学

which is trivial to check.

检查起来很简单。

Example 2

Find the value of [math]\displaystyle{ \sqrt{2}^{ \sqrt{2}^{\sqrt{2}^{\cdots}} } }[/math] where this is done n times (and possibly the interpolated values when n is not an integer). We have $f (x) = 模板:Sqrt x$ . A fixed point is $a = f (2) = 2$ .

Find the value of [math]\displaystyle{ \sqrt{2}^{ \sqrt{2}^{\sqrt{2}^{\cdots}} } }[/math] where this is done n times (and possibly the interpolated values when n is not an integer). We have ^x}}. A fixed point is f(2) 2}}.

找到 < math > sqrt {2} ^ { sqrt {2} ^ { sqrt {2} ^ { cdots }}}} </math > 的值，在这里执行 n 次(当 n 不是整数时可能是插值值)。我们已经有了。一个不动点是 f (2)2}}。

So set x = 1 and $f n (1)$ expanded around the fixed point value of 2 is then an infinite series,

So set x = 1 and expanded around the fixed point value of 2 is then an infinite series,

所以设定 x = 1，并围绕2的不动点展开，就是一个无穷级数,

[math]\displaystyle{ \lt math\gt 《数学》 \sqrt{2}^{ \sqrt{2}^{\sqrt{2}^{\cdots}} } = f^n(1) = 2 - (\ln 2)^n + \frac{(\ln 2)^{n+1}((\ln 2)^n-1)}{4(\ln 2-1)} - \cdots \sqrt{2}^{ \sqrt{2}^{\sqrt{2}^{\cdots}} } = f^n(1) = 2 - (\ln 2)^n + \frac{(\ln 2)^{n+1}((\ln 2)^n-1)}{4(\ln 2-1)} - \cdots Sqrt {2} ^ { sqrt {2} ^ { sqrt {2} ^ { cdots }} = f ^ n (1) = 2-(ln 2) ^ n + frac {(ln 2) ^ { n + 1}((ln 2) ^ n-1)}{4(ln 2-1)}-cdots }[/math]

</math>

数学

which, taking just the first three terms, is correct to the first decimal place when n is positive—cf. Tetration: $f n (1) = n 模板:Sqrt$ . (Using the other fixed point $a = f (4) = 4$ causes the series to diverge.)

which, taking just the first three terms, is correct to the first decimal place when n is positive—cf. Tetration: ⁿ }}. (Using the other fixed point f(4) 4}} causes the series to diverge.)

只取前三项，当 n 是正数时，精确到小数点后第一位。茶滤: n }。(使用另一个不动点 f (4)4}使数列发散)

For $n = -1$ , the series computes the inverse function 模板:Sfrac.

For −1}}, the series computes the inverse function .

对于-1} ，该序列计算反函数。

Example 3

With the function $f (x) = x b$ , expand around the fixed point 1 to get the series

With the function x^b}}, expand around the fixed point 1 to get the series

使用函数 x b } ，在固定点1周围扩展得到序列

[math]\displaystyle{ \lt math\gt 《数学》 f^n(x) = 1 + b^n(x-1) + \frac{1}2b^{n}(b^n-1)(x-1)^2 + \frac{1}{3!}b^n (b^n-1)(b^n-2)(x-1)^3 + \cdots ~, f^n(x) = 1 + b^n(x-1) + \frac{1}2b^{n}(b^n-1)(x-1)^2 + \frac{1}{3!}b^n (b^n-1)(b^n-2)(x-1)^3 + \cdots ~, F ^ n (x) = 1 + b ^ n (x-1) + frac {1}2b ^ { n }(b ^ n-1)(x-1) ^ 2 + frac {1}{3！} b ^ n (b ^ n-1)(b ^ n-2)(x-1) ^ 3 + cdots ~ , }[/math]

</math>

数学

which is simply the Taylor series of x^{(bⁿ )} expanded around 1.

这是简单的泰勒级数 x (b n ) 在1左右膨胀。

Conjugacy

If $f$ and $g$ are two iterated functions, and there exists a homeomorphism $h$ such that $g = h -1 ○ f ○ h$ , then $f$ and $g$ are said to be topologically conjugate.

If and are two iterated functions, and there exists a homeomorphism such that h⁻¹ ○ f ○ h }}, then and are said to be topologically conjugate.

如果和是两个迭代函数，并且存在一个同胚，使得 h -1 f ○ h } ，则和称为拓扑共轭。

Clearly, topological conjugacy is preserved under iteration, as $g n = h -1 ○ f n ○ h$ . Thus, if one can solve for one iterated function system, one also has solutions for all topologically conjugate systems. For example, the tent map is topologically conjugate to the logistic map. As a special case, taking $f (x) = x + 1$ , one has the iteration of $g (x) = h -1 (h (x) + 1)$ as

Clearly, topological conjugacy is preserved under iteration, as h⁻¹ ○ f ⁿ ○ h}}. Thus, if one can solve for one iterated function system, one also has solutions for all topologically conjugate systems. For example, the tent map is topologically conjugate to the logistic map. As a special case, taking x + 1}}, one has the iteration of h⁻¹(h(x) + 1)}} as

很明显，拓扑共轭在迭代过程中被保留了下来，例如 h -1 ○ f n ○ h }。因此，如果一个人可以求出一个迭代函数系统，那么他也可以求出所有拓扑共轭系统的解。例如，帐篷映射拓扑上与 logistic 映射共轭。作为一个特殊情况，取 x + 1} ，有 h & minus; 1 (h (x) + 1)}} As 的迭代

g n (x) = h -1 (h (x) + n)

, for any function

h

.

h⁻¹(h(x) + n)}},     for any function .

H & minus; 1 (h (x) + n)}} ，对于任何函数。

Making the substitution $x = h -1 (y) = ϕ (y)$ yields

Making the substitution h⁻¹(y) ϕ(y)}} yields

取代 h &-; 1 (y)(y)}产额

g (ϕ (y)) = ϕ (y +1)

, a form known as the Abel equation.

ϕ(y+1)}},    a form known as the Abel equation.

(y + 1)} ，一种被称为 Abel 方程的形式。

Even in the absence of a strict homeomorphism, near a fixed point, here taken to be at $x$ = 0, $f$ (0) = 0, one may often solve^[9] Schröder's equation for a function Ψ, which makes $f (x)$ locally conjugate to a mere dilation, $g (x) = f'(0) x$ , that is

Even in the absence of a strict homeomorphism, near a fixed point, here taken to be at = 0, (0) = 0, one may often solve Schröder's equation for a function Ψ, which makes locally conjugate to a mere dilation, f '(0) x}}, that is

即使在没有严格同胚的情况下，在一个不动点附近，这里取为 at = 0，(0) = 0，人们常常可以解一个函数的施罗德方程，它使局部共轭到一个单纯的膨胀，即 f’(0) x }

f (x) = Ψ -1 (f'(0) Ψ(x))

.

 Ψ⁻¹(f '(0) Ψ(x))}}.

 Ψ⁻¹(f '(0) Ψ(x))}}.

Thus, its iteration orbit, or flow, under suitable provisions (e.g., $f'(0) \neq 1$ ), amounts to the conjugate of the orbit of the monomial,

Thus, its iteration orbit, or flow, under suitable provisions (e.g., ), amounts to the conjugate of the orbit of the monomial,

因此，它的迭代轨道，或流，在适当的条件下(例如) ，等于单项式轨道的共轭,

Ψ -1 (f'(0) n Ψ(x))

,

where $n$ in this expression serves as a plain exponent: functional iteration has been reduced to multiplication! Here, however, the exponent $n$ no longer needs be integer or positive, and is a continuous "time" of evolution for the full orbit:^[10] the monoid of the Picard sequence (cf. transformation semigroup) has generalized to a full continuous group.^[11]

where in this expression serves as a plain exponent: functional iteration has been reduced to multiplication! Here, however, the exponent no longer needs be integer or positive, and is a continuous "time" of evolution for the full orbit: the monoid of the Picard sequence (cf. transformation semigroup) has generalized to a full continuous group.

其中在这个表达式作为一个简单的指数: 函数迭代已经减少到乘法！然而，在这里，指数不再需要是整数或正数，而是一个连续的“时间”的演化为完整的轨道: monoid 的 Picard 序列(参见。变换半群)已推广到一个完全连续群。

文件:Sine iterations.svg

Iterates of the sine function (blue), in the first half-period. Half-iterate (orange), i.e., the sine's functional square root; the functional square root of that, the quarter-iterate (black) above it; and further fractional iterates up to the 1/64th. The functions below the (blue) sine are six integral iterates below it, starting with the second iterate (red) and ending with the 64th iterate. The green envelope triangle represents the limiting null iterate, the sawtooth function serving as the starting point leading to the sine function. The dashed line is the negative first iterate, i.e. the inverse of sine (arcsin). Iterates of the sine function (blue), in the first half-period. Half-iterate (orange), i.e., the sine's functional square root; the functional square root of that, the quarter-iterate (black) above it; and further fractional iterates up to the 1/64th. The functions below the (blue) sine are six integral iterates below it, starting with the second iterate (red) and ending with the 64th iterate. The green envelope triangle represents the limiting null iterate, the sawtooth function serving as the starting point leading to the sine function. The dashed line is the negative first iterate, i.e. the inverse of sine (arcsin). 正弦函数( blue )的迭代，在前半段。半迭代( orange ) ，即正弦函数的函数平方根; 其函数平方根，它上面的四分之一迭代(black) ; 以及进一步的小数迭代到1/64。正弦函数( blue )下面有六个整数迭代，从第二个迭代开始( red )到第64个迭代结束。绿色信封三角形代表极限的零迭代，锯齿函数作为正弦函数的起点。虚线是负的第一个迭代，即。正弦的逆(反正弦)。 (From the general pedagogy web-site.^[12] For the notation, see [1].) (From the general pedagogy web-site. For the notation, see [2].) (摘自普通教学法网站。关于这个符号，请参阅[ http://www.physics.miami.edu/~curtright/therootsofsin.pdf ]

]]

This method (perturbative determination of the principal eigenfunction Ψ, cf. Carleman matrix) is equivalent to the algorithm of the preceding section, albeit, in practice, more powerful and systematic.

这种方法(主特征函数的微扰确定法，参数的微扰确定法，参数的微扰确定法，参数的微扰确定法，参数的微扰确定法，参数的微扰确定法，参数的微扰确定法，参数的微扰确定法，参数的微扰确定法。Carleman 矩阵)等价于前面部分的算法，尽管在实践中更加强大和系统化。

Markov chains

If the function is linear and can be described by a stochastic matrix, that is, a matrix whose rows or columns sum to one, then the iterated system is known as a Markov chain.

如果函数是线性的，并且可以用一个转移矩阵矩阵来描述，也就是一个行或列和为一的矩阵，那么这个迭代系统就是马尔可夫链。

Examples

There are many chaotic maps.

这里有许多混乱的地图。

Well-known iterated functions include the Mandelbrot set and iterated function systems.

众所周知的迭代函数包括 Mandelbrot 集和迭代函数系统。

Ernst Schröder,^[13] in 1870, worked out special cases of the logistic map, such as the chaotic case $f (x) = 4 x (1 - x)$ , so that $Ψ(x) = arcsin 2 (模板:Radic)$ , hence $f n (x) = sin 2 (2 n arcsin(模板:Radic))$ .

Ernst Schröder, in 1870, worked out special cases of the logistic map, such as the chaotic case 4x(1 − x)}}, so that arcsin²()}}, hence sin²(2ⁿ arcsin())}}.

Ernst Schröder 在1870年提出了 logistic 映射的特殊情形，如混沌情形4x (1-x)} ，使得 arcsin 2 ()} ，因此 sin 2 (2 n arcsin ())}。

A nonchaotic case Schröder also illustrated with his method, $f (x) = 2 x (1 - x)$ , yielded $Ψ(x) = −模板:Sfrac ln(1 − 2x)$ , and hence $fn(x) = −模板:Sfrac((1 − 2x)2n − 1)$ .

A nonchaotic case Schröder also illustrated with his method, 2x(1 − x)}}, yielded − ln(1 − 2x)}}, and hence −((1 − 2x)^2ⁿ − 1)}}.

对一个非混沌的案例施罗德也给出了他的方法，2x (1-x)} ，产出-ln (1-2x)} ，因此-(1-2x) 2 n -1)}。

If $f$ is the action of a group element on a set, then the iterated function corresponds to a free group.

If is the action of a group element on a set, then the iterated function corresponds to a free group.

如果是一个组元素在一个集合上的动作，那么迭代函数对应于一个自由组。

Most functions do not have explicit general closed-form expressions for the n-th iterate. The table below lists some^[13] that do. Note that all these expressions are valid even for non-integer and negative n, as well as non-negative integer n.

Most functions do not have explicit general closed-form expressions for the n-th iterate. The table below lists some that do. Note that all these expressions are valid even for non-integer and negative n, as well as non-negative integer n.

对于 n 次迭代，大多数函数没有显式的一般闭式表达式。下面的表格列出了一些这样做。注意，所有这些表达式甚至对非整数和负数 n 以及非负整数 n 都有效。

100%

[math]\displaystyle{ f(x) }[/math]	[math]\displaystyle{ f(x) }[/math]	数学 f (x)	[math]\displaystyle{ f^n(x) }[/math]	[math]\displaystyle{ f^n(x) }[/math]	< math > f ^ n (x) </math >
[math]\displaystyle{ x+b }[/math]	[math]\displaystyle{ x+b }[/math]	< math > x + b	[math]\displaystyle{ x+nb }[/math]	[math]\displaystyle{ x+nb }[/math]	< math > x + nb </math >
[math]\displaystyle{ ax+b \ (a\ne 1) }[/math]	[math]\displaystyle{ ax+b \ (a\ne 1) }[/math]	< math > ax + b (a ne 1) </math >	[math]\displaystyle{ a^nx+\frac{a^n-1}{a-1}b }[/math]	[math]\displaystyle{ a^nx+\frac{a^n-1}{a-1}b }[/math]	< math > a ^ nx + frac { a ^ n-1}{ a-1} b </math >
[math]\displaystyle{ ax^b \ (b\ne 1) }[/math]	[math]\displaystyle{ ax^b \ (b\ne 1) }[/math]	< math > ax ^ b (b ne 1) </math >	[math]\displaystyle{ a^{\frac{b^n-1}{b-1}}x^{b^n} }[/math]	[math]\displaystyle{ a^{\frac{b^n-1}{b-1}}x^{b^n} }[/math]	< math > a ^ { frac { b ^ n-1}{ b-1} x ^ { b ^ n } </math >
[math]\displaystyle{ ax^2 + bx + \frac{b^2 - 2b}{4a} }[/math] (see note)	[math]\displaystyle{ ax^2 + bx + \frac{b^2 - 2b}{4a} }[/math] (see note)	< math > ax ^ 2 + bx + frac { b ^ 2-2b }{4a } </math > (见注释) < br >	[math]\displaystyle{ \frac{2\alpha^{2^n} - b}{2a} }[/math]	[math]\displaystyle{ \frac{2\alpha^{2^n} - b}{2a} }[/math]	< math > frac {2 alpha ^ {2 ^ n }-b }{2a } </math > < br > where: where: 在哪里: [math]\displaystyle{ \alpha = \frac{2ax + b}{2} }[/math] [math]\displaystyle{ \alpha = \frac{2ax + b}{2} }[/math] 2ax + b }{2}
[math]\displaystyle{ ax^2 + bx + \frac{b^2 - 2b - 8}{4a} }[/math] (see note)	[math]\displaystyle{ ax^2 + bx + \frac{b^2 - 2b - 8}{4a} }[/math] (see note)	< math > ax ^ 2 + bx + frac { b ^ 2-2b-8}{4a } </math > (见注释) < br >	[math]\displaystyle{ \frac{2\alpha^{2^n} + 2\alpha^{-2^n} - b}{2a} }[/math]	[math]\displaystyle{ \frac{2\alpha^{2^n} + 2\alpha^{-2^n} - b}{2a} }[/math]	< math > frac {2 alpha ^ {2 ^ n } + 2 alpha ^ {-2 ^ n }-b }{2a } </math > < br > where: where: 在哪里: [math]\displaystyle{ \alpha = \frac{2ax + b \pm \sqrt{(2ax + b)^2 - 16}}{4} }[/math] [math]\displaystyle{ \alpha = \frac{2ax + b \pm \sqrt{(2ax + b)^2 - 16}}{4} }[/math] 2ax + b pm sqrt {(2ax + b) ^ 2-16}{4} </math >
[math]\displaystyle{ \frac{ax + b}{cx + d} }[/math] (rational difference equation)^[14]	[math]\displaystyle{ \frac{ax + b}{cx + d} }[/math] (rational difference equation)	< math > frac { ax + b }{ cx + d } </math > (有理差分方程)	[math]\displaystyle{ \frac{a}{c} + \frac{bc - ad}{c} \left [ \frac{(cx - a + \alpha)\alpha^{n - 1} - (cx - a + \beta)\beta^{n - 1}}{(cx - a + \alpha)\alpha^{n} - (cx - a + \beta)\beta^{n}} \right ] }[/math]	[math]\displaystyle{ \frac{a}{c} + \frac{bc - ad}{c} \left [ \frac{(cx - a + \alpha)\alpha^{n - 1} - (cx - a + \beta)\beta^{n - 1}}{(cx - a + \alpha)\alpha^{n} - (cx - a + \beta)\beta^{n}} \right ] }[/math]	< math > frac { a }{ c } + frac { bc-ad }{ c } left [ frac {(cx-a + alpha) alpha ^ { n-1}-(cx-a + beta) beta ^ { n-1}{(cx-a + alpha) alpha ^ { n }-(cx-a + beta) beta ^ { n }-(beta ^ { n }}}右] </math > < br > where: where: 在哪里: [math]\displaystyle{ \alpha = \frac{a + d + \sqrt{(a - d)^2 + 4bc}}{2} }[/math] [math]\displaystyle{ \alpha = \frac{a + d + \sqrt{(a - d)^2 + 4bc}}{2} }[/math] { a + d + sqrt {(a-d) ^ 2 + 4bc }}{2} [math]\displaystyle{ \beta = \frac{a + d - \sqrt{(a - d)^2 + 4bc}}{2} }[/math] [math]\displaystyle{ \beta = \frac{a + d - \sqrt{(a - d)^2 + 4bc}}{2} }[/math] [ math > beta = frac { a + d-sqrt {(a-d) ^ 2 + 4bc }{2} </math >
[math]\displaystyle{ \sqrt{x^2 + b} }[/math]	[math]\displaystyle{ \sqrt{x^2 + b} }[/math]	< math > sqrt { x ^ 2 + b } </math >	[math]\displaystyle{ \sqrt{x^2 + bn} }[/math]	[math]\displaystyle{ \sqrt{x^2 + bn} }[/math]	< math > sqrt { x ^ 2 + bn }
[math]\displaystyle{ \sqrt{ax^2 + b} \ (a \ne 1) }[/math]	[math]\displaystyle{ \sqrt{ax^2 + b} \ (a \ne 1) }[/math]	< math > sqrt { ax ^ 2 + b }(a ne 1) </math >	[math]\displaystyle{ \sqrt{a^nx^2 + \frac{a^n - 1}{a - 1}b} }[/math]	[math]\displaystyle{ \sqrt{a^nx^2 + \frac{a^n - 1}{a - 1}b} }[/math]	< math > sqrt { a ^ nx ^ 2 + frac { a ^ n-1}{ a-1} b } </math >
[math]\displaystyle{ g^{-1}\Big(f\bigl(g(x)\bigr)\Big) }[/math]	[math]\displaystyle{ g^{-1}\Big(f\bigl(g(x)\bigr)\Big) }[/math]	< math > g ^ {-1} Big (f bigl (g (x) bigr) Big) </math >	[math]\displaystyle{ g^{-1}\Bigl(f^n\bigl(g(x)\bigr)\Bigr) }[/math]	[math]\displaystyle{ g^{-1}\Bigl(f^n\bigl(g(x)\bigr)\Bigr) }[/math]	< math > g ^ {-1} Bigl (f ^ n Bigl (g (x) Bigr) </math >
[math]\displaystyle{ g^{-1}\bigl(g(x)+b\bigr) }[/math] (generic Abel equation)	[math]\displaystyle{ g^{-1}\bigl(g(x)+b\bigr) }[/math] (generic Abel equation)	< math > g ^ {-1} bigl (g (x) + b bigr) </math > (通用 Abel 方程)	[math]\displaystyle{ g^{-1}\bigl(g(x)+nb\bigr) }[/math]	[math]\displaystyle{ g^{-1}\bigl(g(x)+nb\bigr) }[/math]	< math > g ^ {-1} bigl (g (x) + nb bigr) </math >
[math]\displaystyle{ g^{-1}\Bigl(a\ g(x)+b\Bigr) \ (a\ne 1) }[/math]	[math]\displaystyle{ g^{-1}\Bigl(a\ g(x)+b\Bigr) \ (a\ne 1) }[/math]	< math > g ^ {-1} Bigl (a g (x) + b Bigr)(a ne 1) </math >	[math]\displaystyle{ g^{-1}\Bigl(a^ng(x)+\frac{a^n-1}{a-1}b\Bigr) }[/math]	[math]\displaystyle{ g^{-1}\Bigl(a^ng(x)+\frac{a^n-1}{a-1}b\Bigr) }[/math]	< math > g ^ {-1} Bigl (a ^ ng (x) + frac { a ^ n-1}{ a-1} b Bigr) </math >
[math]\displaystyle{ T_m (x)=\cos (m \arccos x) }[/math] (Chebyshev polynomial for integer m)	[math]\displaystyle{ T_m (x)=\cos (m \arccos x) }[/math] (Chebyshev polynomial for integer m)	< math > t _ m (x) = cos (m arcos x) </math > (整数 m 的 Chebyshev 多项式)	[math]\displaystyle{ T_{mn}=\cos(mn \arccos x) }[/math]	[math]\displaystyle{ T_{mn}=\cos(mn \arccos x) }[/math]	< math > t _ { mn } = cos (mn arcos x) </math >

|}

Note: these two special cases of $ax 2 + bx + c$ are the only cases that have a closed-form solution. Choosing b = 2 = –a and b = 4 = –a, respectively, further reduces them to the nonchaotic and chaotic logistic cases discussed prior to the table.

Note: these two special cases of are the only cases that have a closed-form solution. Choosing b = 2 = –a and b = 4 = –a, respectively, further reduces them to the nonchaotic and chaotic logistic cases discussed prior to the table.

注意: 这两种特殊情况是唯一具有封闭形式解的情况。选择 b = 2 =-a 和 b = 4 =-a，进一步将它们归结为表前讨论的非混沌和混沌的逻辑情况。

Some of these examples are related among themselves by simple conjugacies. A few further examples, essentially amounting to simple conjugacies of Schröder's examples can be found in ref.^[15]

Some of these examples are related among themselves by simple conjugacies. A few further examples, essentially amounting to simple conjugacies of Schröder's examples can be found in ref.

这些例子中有些是通过简单的配合关系相互联系的。更进一步的例子，本质上相当于施罗德例子的简单变形，可以在参考文献中找到。

Means of study

Iterated functions can be studied with the Artin–Mazur zeta function and with transfer operators.

迭代函数可以用 Artin-Mazur zeta 函数和转移算子来研究。

In computer science

In computer science, iterated functions occur as a special case of recursive functions, which in turn anchor the study of such broad topics as lambda calculus, or narrower ones, such as the denotational semantics of computer programs.

在计算机科学中，迭代函数作为递归函数的一种特殊情况出现，这种特殊情况反过来又锚定了诸如 lambda 微积分这样的广泛主题的研究，或者更窄的主题，诸如计算机程序的指称语义。

Definitions in terms of iterated functions

Two important functionals can be defined in terms of iterated functions. These are summation:

两个重要泛函可以用迭代函数来定义。以下是总结:

[math]\displaystyle{ \lt math\gt 《数学》 \left\{b+1,\sum_{i=a}^b g(i)\right\} \equiv \left( \{i,x\} \rightarrow \{ i+1 ,x+g(i) \}\right)^{b-a+1} \{a,0\} \left\{b+1,\sum_{i=a}^b g(i)\right\} \equiv \left( \{i,x\} \rightarrow \{ i+1 ,x+g(i) \}\right)^{b-a+1} \{a,0\} 左{ b + 1，sum { i = a } ^ b g (i) right } equiv left ({ i，x } right tarrow { i + 1，x + g (i)} right) ^ { b-a + 1}{ a，0} }[/math]

</math>

数学

and the equivalent product:

以及相应的产品:

[math]\displaystyle{ \lt math\gt 《数学》 \left\{b+1,\prod_{i=a}^b g(i)\right\} \equiv \left( \{i,x\} \rightarrow \{ i+1 ,x g(i) \}\right)^{b-a+1} \{a,1\} \left\{b+1,\prod_{i=a}^b g(i)\right\} \equiv \left( \{i,x\} \rightarrow \{ i+1 ,x g(i) \}\right)^{b-a+1} \{a,1\} 左{ b + 1，prod _ { i = a } ^ b (i) right }等价左({ i，x } right tarrow { i + 1，x g (i)}右) ^ { b-a + 1}{ a，1} }[/math]

</math>

数学

Functional derivative

The functional derivative of an iterated function is given by the recursive formula:

泛函导数的迭代函数是由递归公式给出的:

[math]\displaystyle{ \frac{ \delta f^N(x)}{\delta f(y)} = f'( f^{N-1}(x) ) \frac{ \delta f^{N-1}(x)}{\delta f(y)} + \delta( f^{N-1}(x) - y ) }[/math]

{ delta f ^ n (x)}{ delta f (y)} = f’(f ^ { N-1}(x))) frac { delta f ^ { N-1}(x)}{ delta f (y)} + delta (f ^ { N-1}(x)-y) </math >

Lie's data transport equation

Iterated functions crop up in the series expansion of combined functions, such as $g (f (x))$ .

Iterated functions crop up in the series expansion of combined functions, such as .

迭代函数在组合函数的级数展开中出现，例如。

Given the iteration velocity, or beta function (physics),

给定迭代速度或贝塔函数(物理学) ,

[math]\displaystyle{ v(x) = \left. \frac{\partial f^n(x)}{\partial n} \right|_{n=0} }[/math]

= math > v (x) = left.{ partial f ^ n (x)}{ partial n } right | { n = 0} </math >

for the $n$ ^th iterate of the function $f$ , we have^[16]

for the ^th iterate of the function , we have

对于函数的迭代，我们有

[math]\displaystyle{ \lt math\gt 《数学》 g(f(x)) = \exp\left[ v(x) \frac{\partial}{\partial x} \right] g(x). g(f(x)) = \exp\left[ v(x) \frac{\partial}{\partial x} \right] g(x). G (f (x)) = exp left [ v (x) frac { partial }{ partial x } right ] g (x). }[/math]

</math>

数学

For example, for rigid advection, if $f (x) = x + t$ , then $v (x) = t$ . Consequently, $g (x + t) = exp(t \partial/\partial x) g (x)$ , action by a plain shift operator.

For example, for rigid advection, if x + t}}, then t}}. Consequently, exp(t ∂/∂x) g(x)}}, action by a plain shift operator.

例如，对于刚性对流，如果 x + t } ，那么 t }}。因此，exp (t/x) g (x)} ，由普通的移位操作符操作。

Conversely, one may specify $f (x)$ given an arbitrary $v (x)$ , through the generic Abel equation discussed above,

Conversely, one may specify given an arbitrary , through the generic Abel equation discussed above,

反过来，人们可以通过上面讨论的通用 Abel 方程来指定给定的任意一个,

[math]\displaystyle{ \lt math\gt 《数学》 f(x) = h^{-1}(h(x)+1) , f(x) = h^{-1}(h(x)+1) , F (x) = h ^ {-1}(h (x) + 1) , }[/math]

</math>

数学

where

在哪里

[math]\displaystyle{ \lt math\gt 《数学》 h(x) = \int \frac{1}{v(x)} \, dx . h(x) = \int \frac{1}{v(x)} \, dx . H (x) = int frac {1}{ v (x)} ，dx. }[/math]

</math>

数学

This is evident by noting that

这一点很明显

[math]\displaystyle{ f^n(x)=h^{-1}(h(x)+n)~. }[/math]

< math > f ^ n (x) = h ^ {-1}(h (x) + n) ~ . </math >

For continuous iteration index $t$ , then, now written as a subscript, this amounts to Lie's celebrated exponential realization of a continuous group,

For continuous iteration index , then, now written as a subscript, this amounts to Lie's celebrated exponential realization of a continuous group,

对于连续迭代索引，现在写作下标，这相当于李著名的连续群的指数实现,

[math]\displaystyle{ e^{t~\frac{\partial ~~}{\partial h(x)}} g(x)= g(h^{-1}(h(x )+t))= g(f_t(x)). }[/math]

(x) = g (h ^ {-1}(h (x) + t)) = g (f _ t (x)) . </math >

The initial flow velocity $v$ suffices to determine the entire flow, given this exponential realization which automatically provides the general solution to the translation functional equation,引用错误：没有找到与</ref>对应的<ref>标签

Equations and Their Applications (Dover Books on Mathematics, 2006), Ch. 6, .</ref>

方程及其应用(多佛数学书籍，2006)。6，. </ref >

[math]\displaystyle{ f_t(f_\tau (x))=f_{t+\tau} (x) ~. }[/math]

< math > f _ t (f _ tau (x)) = f _ { t + tau }(x) ~ . </math >

References

↑ Kuczma, Marek (1968). Functional equations in a single variable. Monografie Matematyczne. Warszawa: PWN – Polish Scientific Publishers.
↑ Kuczma, M., Choczewski B., and Ger, R. (1990). Iterative Functional Equations. Cambridge University Press. ISBN 0-521-35561-3. https://archive.org/details/iterativefunctio0000kucz.
↑ Carleson, L.; Gamelin, T. D. W. (1993). Complex dynamics. Universitext: Tracts in Mathematics. Springer-Verlag. ISBN 0-387-97942-5. https://archive.org/details/complexdynamics0000carl.
↑ Istratescu, Vasile (1981). Fixed Point Theory, An Introduction, D. Reidel, Holland. .
↑ "Finding f such that f(f(x))=g(x) given g". MathOverflow.
↑ Aldrovandi, R.; Freitas, L. P. (1998). "Continuous Iteration of Dynamical Maps". J. Math. Phys. 39 (10): 5324. arXiv:physics/9712026. Bibcode:1998JMP....39.5324A. doi:10.1063/1.532574. hdl:11449/65519.
↑ Berkolaiko, G.; Rabinovich, S.; Havlin, S. (1998). "Analysis of Carleman Representation of Analytical Recursions". J. Math. Anal. Appl. 224: 81–90. doi:10.1006/jmaa.1998.5986.
↑ "Tetration.org".
↑ Kimura, Tosihusa (1971). "On the Iteration of Analytic Functions", Funkcialaj Ekvacioj 14, 197-238.
↑ Curtright, T.L.; Zachos, C.K. (2009). "Evolution Profiles and Functional Equations". Journal of Physics A. 42 (48): 485208. arXiv:0909.2424. Bibcode:2009JPhA...42V5208C. doi:10.1088/1751-8113/42/48/485208.
↑ For explicit instance, example 2 above amounts to just $f n (x) = Ψ -1 ((ln 2) n Ψ(x))$ , for any n, not necessarily integer, where Ψ is the solution of the relevant Schröder's equation, $Ψ(模板:Sqrt x) = ln 2 Ψ(x)$ . This solution is also the infinite m limit of $(f m (x) - 2)/(ln 2) m$ .
↑ Curtright, T.L. Evolution surfaces and Schröder functional methods.
↑ ^13.0 ^13.1 Schröder, Ernst (1870). "Ueber iterirte Functionen". Math. Ann. 3 (2): 296–322. doi:10.1007/BF01443992.
↑ Brand, Louis, "A sequence defined by a difference equation," American Mathematical Monthly 62, September 1955, 489–492. online
↑ Katsura, S.; Fukuda, W. (1985). "Exactly solvable models showing chaotic behavior". Physica A: Statistical Mechanics and Its Applications. 130 (3): 597. Bibcode:1985PhyA..130..597K. doi:10.1016/0378-4371(85)90048-2.
↑ Berkson, E.; Porta, H. (1978). "Semigroups of analytic functions and composition operators". The Michigan Mathematical Journal. 25: 101–115. doi:10.1307/mmj/1029002009. Curtright, T. L.; Zachos, C. K. (2010). "Chaotic maps, Hamiltonian flows and holographic methods". Journal of Physics A: Mathematical and Theoretical. 43 (44): 445101. arXiv:1002.0104. Bibcode:2010JPhA...43R5101C. doi:10.1088/1751-8113/43/44/445101.

Category:Dynamical systems

类别: 动力系统

Category:Fractals

分类: 分形

Category:Sequences and series

类别: 序列和系列

Category:Fixed points (mathematics)

类别: 定点(数学)

Category:Functions and mappings

类别: 函数和映射

Category:Functional equations

类别: 功能方程

This page was moved from wikipedia:en:Iterated function. Its edit history can be viewed at 迭代函数/edithistory

[1] Kuczma, Marek (1968). Functional equations in a single variable. Monografie Matematyczne. Warszawa: PWN – Polish Scientific Publishers.

[2] Kuczma, M., Choczewski B., and Ger, R. (1990). Iterative Functional Equations. Cambridge University Press. ISBN 0-521-35561-3. https://archive.org/details/iterativefunctio0000kucz.

[3] Carleson, L.; Gamelin, T. D. W. (1993). Complex dynamics. Universitext: Tracts in Mathematics. Springer-Verlag. ISBN 0-387-97942-5. https://archive.org/details/complexdynamics0000carl.

[4] Istratescu, Vasile (1981). Fixed Point Theory, An Introduction, D. Reidel, Holland. .

[5] "Finding f such that f(f(x))=g(x) given g". MathOverflow.

[6] Aldrovandi, R.; Freitas, L. P. (1998). "Continuous Iteration of Dynamical Maps". J. Math. Phys. 39 (10): 5324. arXiv:physics/9712026. Bibcode:1998JMP....39.5324A. doi:10.1063/1.532574. hdl:11449/65519.

[7] Berkolaiko, G.; Rabinovich, S.; Havlin, S. (1998). "Analysis of Carleman Representation of Analytical Recursions". J. Math. Anal. Appl. 224: 81–90. doi:10.1006/jmaa.1998.5986.

[8] "Tetration.org".

[9] Kimura, Tosihusa (1971). "On the Iteration of Analytic Functions", Funkcialaj Ekvacioj 14, 197-238.

[10] Curtright, T.L.; Zachos, C.K. (2009). "Evolution Profiles and Functional Equations". Journal of Physics A. 42 (48): 485208. arXiv:0909.2424. Bibcode:2009JPhA...42V5208C. doi:10.1088/1751-8113/42/48/485208.

[11] For explicit instance, example 2 above amounts to just $f n (x) = Ψ -1 ((ln 2) n Ψ(x))$ , for any n, not necessarily integer, where Ψ is the solution of the relevant Schröder's equation, $Ψ(模板:Sqrt x) = ln 2 Ψ(x)$ . This solution is also the infinite m limit of $(f m (x) - 2)/(ln 2) m$ .

[12] Curtright, T.L. Evolution surfaces and Schröder functional methods.

[schr-13] 13.0 ^13.1 Schröder, Ernst (1870). "Ueber iterirte Functionen". Math. Ann. 3 (2): 296–322. doi:10.1007/BF01443992.

[14] Brand, Louis, "A sequence defined by a difference equation," American Mathematical Monthly 62, September 1955, 489–492. online

[15] Katsura, S.; Fukuda, W. (1985). "Exactly solvable models showing chaotic behavior". Physica A: Statistical Mechanics and Its Applications. 130 (3): 597. Bibcode:1985PhyA..130..597K. doi:10.1016/0378-4371(85)90048-2.

[16] Berkson, E.; Porta, H. (1978). "Semigroups of analytic functions and composition operators". The Michigan Mathematical Journal. 25: 101–115. doi:10.1307/mmj/1029002009. Curtright, T. L.; Zachos, C. K. (2010). "Chaotic maps, Hamiltonian flows and holographic methods". Journal of Physics A: Mathematical and Theoretical. 43 (44): 445101. arXiv:1002.0104. Bibcode:2010JPhA...43R5101C. doi:10.1088/1751-8113/43/44/445101.

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迭代函数

目录

Definition

Abelian property and iteration sequences

Fixed points

Limiting behaviour

Invariant measure

Fractional iterates and flows, and negative iterates

Some formulas for fractional iteration

Example 1

Example 2

Example 3

Conjugacy

Markov chains

Examples

Means of study

In computer science

Definitions in terms of iterated functions

Functional derivative

Lie's data transport equation

See also

References