Measure-Preserving Transformations

Check that the following transformations preserve Lebesgue measure:

Check that the transformation

(the ``baker's transformation'') preserves Lebesgue measure $\lambda_{2}$.

Let $X=[0,1) \times [0,1), A=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)$ be an integer matrix with $\operatorname {det} A= \pm 1$. Define the transformation $T_{A}: X \rightarrow X$ by the formulas $T_{A}\left(x_{1}, x_{2}\right)=\left(y_{1}, y_{2}\right)$ for $\left(x_{1}, x_{2}\right) \in X$, where

Prove that

Let the map $T_{A}: X \rightarrow X$ be the same as in the previous problem, but with $\operatorname {det} A \in \mathbb {Z},\left|\operatorname {det} A\right|>1$. Such a map (no longer one-to-one) is called an endomorphism of the torus. Prove that it preserves Lebesgue measure.

Check that the map $T(x)=\frac{1}{x} \bmod 1(x \in (0,1])$ preserves the measure $d \mu =\frac{C}{1+x} d x$.

Find an absolutely continuous invariant measure

Prove that a transformation $T$ has an invariant (respectively, finite invariant, absolutely continuous invariant) measure if and only if some power $T^{n}(n \in \mathbb {N})$ of it has this property.

Let $T:\left[a, b\right] \rightarrow [a, b]$ be a strictly monotone continuous transformation. Prove that it has an absolutely continuous invariant measure (in general, an infinite one).

Let the transformation $T:\left[a, b\right] \rightarrow [a, b]$ be piecewise monotone and smooth. Prove that if it has an attracting cycle (see 1), then $T$ does not preserve any finite measure equivalent to Lebesgue measure.

Does there exist a nonzero measure on $\mathbb {R}$ invariant with respect to all

Does there exist a nontrivial measure on $(0,+\infty )$ invariant with respect to all transformations of the form

A measure $\mu$ on $\mathbb {R}^{n}$ is called quasi-invariant with respect to shifts if for every $a \in \mathbb {R}^{n}$ the condition $\mu (A)=0$ is equivalent to the condition $\mu (A+a)=0$.

Prove that

Let $T$ be a transformation of a set $X \subset \mathbb {R}^{n}$ into itself such that the image $T \lambda_{n}$ of the measure $\lambda_{n}$ is absolutely continuous with respect to $\lambda_{n}$, and let $f$ be an arbitrary summable function on $X$. Prove the following statements:

Let $n \in \mathbb {N}, \Delta_{i}=\left[\frac{i-1}{n}, \frac{i}{n}\right)(i=1, \ldots , n)$. Consider the nonnegative functions on $[0,1)$ that are constant on the intervals $\Delta_{i}$. Each such function is given by a vector $c=\left(c_{1}, \ldots , c_{n}\right)$ of the values it takes on $\Delta_{1}, \ldots , \Delta_{n}$. For clarity, let us denote such a function by the symbol $f_{c}$. Denote the set of nonnegative vectors in $\mathbb {R}^{n}$ by $\mathbb {R}_{+}^{n}$.

Let $T:[0,1) \rightarrow [0,1)$ be an arbitrary transformation.

Let $n \in \mathbb {N}, \Delta_{i}=\left[\frac{i-1}{n}, \frac{i}{n}\right)(i=1, \ldots , n)$. A nonnegative function on $[0,1)$ that is constant on the intervals $\Delta_{i}$ and takes the values $c_{1}, \ldots , c_{n}$ on $\Delta_{1}, \ldots , \Delta_{n}$ is denoted by $f_{c}$, where $c=\left(c_{1}, \ldots , c_{n}\right)$.

Let $T:\left[0,1\right] \rightarrow [0,1]$.

Let $\Omega =\left\{ z \in \mathbb {C} \mid \operatorname {Im} z>0\right\}$. Find an absolutely continuous measure invariant under all conformal automorphisms of $\Omega$, i.e., transformations $T(z)=\frac{a z+b}{c z+d}$, where $a, b, c, d \in \mathbb {R}$ and $\operatorname {det}\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)>0$.

Let $P=\left\{ (x, y) \in \mathbb {R}^{2} \mid x>0\right\}$. For each point $(a, b) \in P$ define the transformations $L_{a, b}$ and $R_{a, b}$ of the half-plane $P$ into itself:

Find absolutely continuous measures $\mu_{L}$ and $\mu_{R}$ on $P$ invariant under all transformations $L_{a, b}$ and $R_{a, b}$, respectively.

Note that the transformations in question correspond to the left and right shift transformations on the group (under multiplication) of matrices

i.e., to the transformations $M_{x, y} \mapsto M_{a, b} M_{x, y} ; M_{x, y} \mapsto M_{x, y} M_{a, b}$. The measures $\mu_{L}$ and $\mu_{R}$ are called the Haar measures (left and right) for this group.

Prove that if a transformation $T:\left[a, b\right] \rightarrow [a, b]$ preserves some finite absolutely continuous measure, then $T$ is topologically conjugate to some transformation of the same interval preserving Lebesgue measure.

Prove that if $T$ is a homeomorphism of the circle $S^{1}$ with irrational rotation number $\alpha$ (see problem 1.43), then $T$ is related to the rotation $T_{\alpha }$ of the circle by the angle $\alpha$ (i.e., $\left.T_{\alpha }\left(e^{2 \pi i x}\right)=e^{2 \pi i(x+\alpha )}\right)$ as follows: $h \circ T=T_{\alpha } \circ h$ for some continuous mapping $h: S^{1} \rightarrow S^{1}$.

The measure $\mu$ is assumed to be finite and to satisfy the normalization condition $\mu (X)=1$ (such measures are called probability measures). Let $T$ be a transformation of the set $X$. A set $A \subset X$ is called invariant if $T^{-1}(A)=A$. A transformation $T$ preserving the measure $\mu$ is called ergodic if for every invariant set $A \subset X$ one of the following two holds: either $\mu (A)=0$, or $\mu (X \backslash A)=0$.

A measurable function $f$ defined on a set $X$ with measure $\mu$ is called invariant under a transformation $T: X \rightarrow X$ if $f(T x)=f(x)$ for almost all $x$. Prove that for $T$ to be ergodic it is necessary and sufficient that every invariant measurable function coincide with a constant almost everywhere.

The measure $\mu$ is assumed to be finite and to satisfy the normalization condition $\mu (X)=1$ (such measures are called probability measures). Let $T$ be a transformation of the set $X$. A set $A \subset X$ is called invariant if $T^{-1}(A)=A$. A transformation $T$ preserving the measure $\mu$ is called ergodic if for every invariant set $A \subset X$ one of the following two holds: either $\mu (A)=0$, or $\mu (X \backslash A)=0$.

Are the following transformations ergodic:

The measure $\mu$ is assumed to be finite and to satisfy the normalization condition $\mu (X)=1$ (such measures are called probability measures). Let $T$ be a transformation of the set $X$. A set $A \subset X$ is called invariant if $T^{-1}(A)=A$. A transformation $T$ preserving the measure $\mu$ is called ergodic if for every invariant set $A \subset X$ one of the following two holds: either $\mu (A)=0$, or $\mu (X \backslash A)=0$.

Let $X$ be a set in $\mathbb {R}^{n}$ that is the closure of the set of its interior points, $T$ a transformation of $X$ preserving Lebesgue measure. Prove that if $T$ is ergodic, then for almost every $x \in X$ the trajectory of the point $x$ is everywhere dense.

The measure $\mu$ is assumed to be finite and to satisfy the normalization condition $\mu (X)=1$ (such measures are called probability measures). Let $T$ be a transformation of the set $X$. A set $A \subset X$ is called invariant if $T^{-1}(A)=A$. A transformation $T$ preserving the measure $\mu$ is called ergodic if for every invariant set $A \subset X$ one of the following two holds: either $\mu (A)=0$, or $\mu (X \backslash A)=0$.

Let $X=[0,1)^{n}$ be the $n$-dimensional torus $(n \in \mathbb {N}), S$ the shift $x \mapsto y=S(x), \quad$ given by the formula $\quad y_{k}=\left(x_{k}+\alpha_{k}\right) \bmod 1$, $k=1, \ldots , n$. For which values of $\alpha_{1}, \ldots , \alpha_{n}$ is the shift ergodic?

The measures are assumed to be finite and normalized (probability measures): $\mu (X)=1$.

Prove that the maps of the segment $\left[0,1\right]$ into itself given by the formulas

are metrically conjugate. (Maps $T: X \rightarrow X$ and $S: Y \rightarrow Y$ preserving measures $\mu$ and $v$ respectively are called metrically conjugate if there exists a measurable map $h: X \rightarrow Y$ such that $h \mu =v$ and $S \circ h(x)=h \circ T(x)$ for almost all $x \in X$.)

The measure $\mu$ is assumed to be finite and to satisfy the normalization condition $\mu (X)=1$ (such measures are called probability measures). Let $T$ be a transformation of the set $X$. A set $A \subset X$ is called invariant if $T^{-1}(A)=A$. A transformation $T$ preserving the measure $\mu$ is called ergodic if for every invariant set $A \subset X$ one of the following two holds: either $\mu (A)=0$, or $\mu (X \backslash A)=0$.

Prove that the map $T$ of the segment $\left[-1,1\right]$ into itself, preserving the measure $\mu$, is ergodic if

A measure $\mu$ is called a probability measure if it is finite and $\mu (X)=1$.

Prove that if $T: X \rightarrow X$ is a transformation preserving a probability measure $\mu$, and $f$ is a positive function, then $\sum_{k} f\left(T^{k}(x)\right)=+\infty$ for almost all (with respect to $\mu$) $x \in X$.

The measure $\mu$ is assumed to be finite and to satisfy the normalization condition $\mu (X)=1$ (such measures are called probability measures). Let $T$ be a transformation of the set $X$. A set $A \subset X$ is called invariant if $T^{-1}(A)=A$. A transformation $T$ preserving the measure $\mu$ is called ergodic if for every invariant set $A \subset X$ one of the following two holds: either $\mu (A)=0$, or $\mu (X \backslash A)=0$.

Let $T: X \rightarrow X$ be an ergodic transformation preserving a probability measure $\mu , A$ a measurable subset of $X$.

Let $\varepsilon_{n}=0$ if $T^{n}(x) \notin A$, and $\varepsilon_{n}=1$ if $T^{n}(x) \in A(n \geqslant 0)$. Prove that for almost all $x \in X$ the limit $\lim \frac{1}{n} \sum_{0 \leqslant k<n} \varepsilon_{k}$ exists (the average number of hits of the trajectory into the set $A$), and find it.

A measure $\mu$ is called a probability measure if it is finite and $\mu (X)=1$. Let $T$ be a transformation of the set $X$. A set $A \subset X$ is called invariant if $T^{-1}(A)=A$. A transformation $T$ preserving the measure $\mu$ is called ergodic if for every invariant set $A \subset X$ one of the following two holds: either $\mu (A)=0$, or $\mu (X \backslash A)=0$. Probability measures $\mu_{1}, \mu_{2}$ on $X$ are called mutually singular if there exists a set $A \subset X$ such that $\mu_{1}(A)=1$ and $\mu_{2}(A)=0$.

Let $\mu_{1}$ and $\mu_{2}$ be two probability measures on $X, T$ a transformation ergodic with respect to both $\mu_{1}$ and $\mu_{2}$. Prove that the measures $\mu_{1}$ and $\mu_{2}$ either coincide or are mutually singular.

The measure $\mu$ is assumed to be finite and to satisfy the normalization condition $\mu (X)=1$ (such measures are called probability measures). Let $T$ be a transformation of the set $X$. A set $A \subset X$ is called invariant if $T^{-1}(A)=A$. A transformation $T$ preserving the measure $\mu$ is called ergodic if for every invariant set $A \subset X$ one of the following two holds: either $\mu (A)=0$, or $\mu (X \backslash A)=0$.

Consider the irrational shift

It is ergodic by 2.21a). By the ergodic theorem, for any summable function $f$ on $X$ and for almost all $x \in X$ the equality

holds.

Let $x=\sum \frac{x_{k}}{2^{k}}, x_{k} \in \left\{ 0,1\right\}$ be the binary expansion of a number $x \in [0,1)$. Using the ergodicity of the doubling transformation (see problem 2.21b)), prove that for almost all $x \in [0,1)$ the limit $\lim \frac{1}{n} \sum_{0 \leqslant k<n} x_{k}$ exists (the frequency of occurrence of the digit one), and find it. Solve the analogous problem for expansions in a base-$p$ fraction (cf. problems IX.2.14, IX.2.16).

Prove that if a transformation $T: X \rightarrow X$ preserving a probability measure $\mu$ has the property

(the mixing property with respect to the measure $\mu$), then $T$ is ergodic.

Prove that the transformation

is mixing, while the irrational rotation (see problem 2.29) is not mixing with respect to Lebesgue measure.

Let $p \in (0,1), q=1-p, \mu_{p}$ be the measure on the interval $\left[0,1\right]$ defined as follows: if $\varepsilon_{1}, \ldots , \varepsilon_{n} \in \left\{ 0,1\right\}$ and

(here $x=\sum \frac{x_{k}}{2^{k}}$ is the binary expansion of the number $x$), then

and the measure is extended to the remaining intervals by additivity. Verify that

1. the transformation $T(x)=(2 x) \bmod 1$ is mixing with respect to the measure $\mu_{p}$ (see problem 2.32);

2. $\mu_{1 / 2}=\lambda$;

3. the measures $\mu_{p}$ and $\lambda$ are mutually singular for $p \neq \frac{1}{2}$;

4. $\mu_{p_{1}}$ is mutually singular with $\mu_{p_{2}}$ for $p_{1} \neq p_{2}$.

Let $a_{k}$ be the first digit of the decimal representation of the number $2^{k} (k=0,1, \ldots )$, i.e., $a_{0}=1, a_{1}=2, a_{2}=4, a_{3}=8, a_{4}=1, \ldots$ What is the frequency of occurrence of the digit $p \in \left\{ 1, \ldots , 9\right\}$ in this sequence?

As is well known, every number $x$ from the interval $(0,1]$ can be uniquely represented as a continued fraction

which is also written as $x=\left[a_{1}, a_{2}, \ldots \right]$, where $a_{1}=a_{1}(x), a_{2}=a_{2}(x)$, $a_{3}=a_{3}(x), \ldots$ are natural numbers. Such a fraction contains finitely many terms only when $x$ is rational. The finite continued fraction $\left[a_{1}, \ldots , a_{n}\right]$ is uniquely written as an irreducible fraction $\frac{p_{n}}{q_{n}}$. The value of an infinite continued fraction is understood as the limit of the finite ones:

the number $\frac{p_{n}}{q_{n}}=\frac{p_{n}(x)}{q_{n}(x)}$ is called the convergent. It is easily verified that

A variety of information about continued fractions can be gleaned from the book $[\mathrm{X}]$. Associated with the expansion of numbers into continued fractions is the Gauss transformation

Indeed, let $x=\left[a_{1}, a_{2}, \ldots \right], y=\left[a_{2}, a_{3}, \ldots \right]$. Then $x=\frac{1}{a_{1}+y}$, from which it follows that $y=T(x)$. Using $T$, the numbers $a_{n}$ are written as follows:

The Gauss transformation preserves the measure $d \mu =\frac{c}{1+x} d x$ (see problem 2.5). The measure $\mu$ is a probability measure for $c=\frac{1}{\ln 2}$. The reader can learn about interesting applications of ergodic theory to number theory and continued fractions in [Po].