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I am trying to prove the following result:

Let $\left( a, b \right) \subseteq \mathbb{R}$ be an interval and $x, y \in \left( a, b \right)$. Then, there is a smooth family of diffeomorphisms $f_t: \left( a, b \right) \rightarrow \left( a, b \right)$, where $0 \leq t \leq 1$, such that the following hold:

  1. For all $z \in \left( a, b \right)$, we have $f_0 \left( z \right) = z$.
  2. There is some $\epsilon > 0$ such that for all $z \in \left( a, a + \epsilon \right) \cup \left( b - \epsilon, b \right)$, we have $f_t \left( z \right) = z$, for all $0 \leq t \leq 1$.
  3. $f_1 \left( x \right) = y$.

I have tried to define $f_t: \left( a, b \right) \rightarrow \left( a, b \right)$ as

$$f_t \left( z \right) = z + t \int\limits_{a}^{z} g \left( u \right) \mathrm{d}u,$$

for some smooth map $g: \left( a, b \right) \rightarrow \left( a, b \right)$. My choice of the smooth map $g = g_1 - g_2$, where $g_1$ and $g_2$ are "bump functions" with (compact) support contained in $\left( a, b \right)$ about $x$ and $y$ such that their areas are same (and they don't overlap), That way, point (1) and (2) of the result are taken care of.

The problem is with point (3) and the bijectivity of $f_t$ (and, indeed, finding its inverse).

For point (3) to hold, we would require

$$\int\limits_{a}^{x} g \left( u \right) \mathrm{d}u = y - x.$$

Also, proving bijectivity with this choice of $g$ is difficult.

So, the question is whether such choice of bump functions $g_1$ and $g_2$ is possible?

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  • $\begingroup$ The third requirement $f_{1}(x)=y$ does not really make sense unless we understand $y$ as a function of $x$. $\endgroup$
    – Medo
    Commented Aug 14, 2021 at 6:59

1 Answer 1

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Using your notations and in particular, we choose $\epsilon >0$ so that $$\tag{1} b-x - 4\epsilon > y-x$$ and $g_1, g_2$ are non-negative,

  • $g_1$ is supported in $(a+\epsilon, x-\epsilon)$,
  • $g_2$ is supported in $(x+\epsilon, b-\epsilon)$,
  • $\int_a^b g_1(u)du = \int_a^b g_2(u)du = y-x$, and

$$\tag{2} g_2 (u) <1,\ \ \ \forall u\in (a, b).$$

We can assume this last condition because of (1). Using $g$, $f_t$ as suggested, we have

$$f_1 (x) = x + \int_a^x g(u) du = x + \int_a^x g_1 (u) du = x + y-x = y.$$

also, by (2),

$$ f_t'(z) = 1 + g(z) = (1 -tg_2(z)) + tg_1(z) >0.$$ Thus each $f_t$ is strictly increasing and thus is injective. Intermediate value theorem imply that $f_t$ is also surjective. Thus $f_t$ is bijective for all $t\in [0,1]$.

Remark To construct $g_2$ with the required property:

  • Consider the function $h : (a, b)\to \mathbb R$ given by $$ h(z) = \begin{cases} \frac{y-x}{b-x-4\epsilon} & \text{ when } z\in [x+ 2\epsilon, b-2\epsilon]\\ 0 & \text{ otherwise.}\end{cases}$$ then $h(z) <1$ by (1) and $\int_a^b h(s) ds = y-x$.
  • Let $\psi$ be a smooth bump function with integral one and support in $(-\epsilon, \epsilon)$. Let $g_2 = h *\psi$ be the convolution. Then $g_2$ is smooth, supported in $[x+ \epsilon, b-\epsilon]$, satisfies (2) and $\int_a^b g_2(s) ds = y-x$ (the last condition follows from the property about convulution (see here))
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  • $\begingroup$ This does take away the question of bijectivity (and $f_t$ being a diffeomorphism). However, the main question is whether we can always choose functions $g_1$ and $g_2$ satisfying the integral conditions? $\endgroup$ Commented Aug 14, 2021 at 10:55
  • $\begingroup$ I have added something about how to construct $g_2$. @AniruddhaDeshmukh $\endgroup$ Commented Aug 14, 2021 at 15:04
  • $\begingroup$ Your definition of $h$ seems to be constant on the interval $\left[ x + 2 \epsilon, b - 2 \epsilon \right]$. Did you mean to write some other variable instead of $x$? $\endgroup$ Commented Aug 22, 2021 at 12:12
  • $\begingroup$ @AniruddhaDeshmukh I have made an edit. The variable should not be $x$ $\endgroup$ Commented Aug 22, 2021 at 13:46
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    $\begingroup$ @AniruddhaDeshmukh It is easier to construct $g_1$, since I don't need any bound on it. Just take $g_1$ to be any non-negative smooth function supported in $(a+\epsilon, x-\epsilon)$ with $\int g_1 = y-x$. $\endgroup$ Commented Aug 23, 2021 at 18:32

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