none

Mhm makes sense, that's what I was expecting, thanks!

But for some reason it's bothering me.

It's in a textbook so we most likely can do that.

The $g_i$ are restrictions of the $f_i$ to the subspace. I can post the full statement of the theorem if you want.

Suppose we have a finite dimensional vector space $V$ and we're proving some statement about linear functionals, $\{f_1, \ldots, f_k \}$. We want to induct on the number of functionals, $k$. Now consider a subspace $W \subset V$ and suppose $\{g_1, \ldots, g_{k-1} \}$ satisfy whatever conditions we need, can we apply the induction hypothesis to the $g_{i}$? The main difference is that we assumed the hypothesis for the space V.

But technically $\rho(f_1) = 1 = 0 \mod 1$.

I was wondering whether there is any good justification for giving a circle map rotation number $1$ as opposed to $0$ (the rotation number is taken mod $1$). There are cases in which it seems to make sense to do so. For instance, considering a continuously varying family of circle maps starting with $\rho(f_0) = 0$ and ending with $\rho(f_1) = 1$. Then every rotation number in between is attained, by continuity of $\rho$.

Is anyone here familiar with rotation numbers?

(and for my purposes this will be enough for now)

Also, I think I figured out how to guarantee an interval around a number such that at least the first number of the continued fraction expansion coincides

Thank you, this seems very nice (and there's a chapter on decimal/continued fraction relationships).

but yes thank you for pointing that out, the algorithm seems helpful

Ah sorry you emphasized the word UNIQUE I thought you were implying it followed from that. I see now that the algorithm for computing them can give a lot of insight, but it's still not that easy I think. At each step of the algorithm we want to get the integer part of performing some division, so the question translates to how these division are affected by increasing # of decimal digits

@rapasite I don't think this follows that directly from uniqueness, but I'll spend some time right now thinking about it and I'll come back with observations/comments.

Doing it for the golden ratio would actually be perfect, I'll try seeing if things are easier in that specific case

does that answer my question? Maybe it does, but I don't quite see it.

not just approximating by evaluating the continued fraction

I specifically want the statement about the terms of the continued fraction itself

Given $n$ and irrational $x \in [0,1]$, does there exist an $\epsilon$ such that for all $|x-y| < \epsilon$, the first $n$ terms of the continued fraction expansions of $x$ and $y$ coincide?

Maybe what I'm saying turns out to be false but I certainly hope not

I can't rely on a probabilistic/average result though, I need to guarantee that at some decimal precision, we obtain some number of terms of the continued fraction that agree. I'm willing to require huge decimal precision for even a few terms of the continued fraction to agree.

Here is the theorem I had in mind: en.wikipedia.org/wiki/Lochs%27s_theorem

@schn if you're inside the unit circle, then $x^2 + y^2 < 1$, so the denominator is negative. If you're outside the the denominator is positive.

the idea stills seems more or less correct, the number of $1$'s will increase as we get closer to the $0.618033 \ldots$, but it's not completely correct that adding extra decimal digits only extends the continued fraction

Type in 0.61 in the decimal section and look at the continued fraction, then type in 0.618 and check it again

But that's not true, I just checked an online calculator and adding extra digits does change the last few terms

Ah I guess the thing to note would be that adding extra decimal digits would only add terms to the continued fraction, not alter the previous terms

the closeness in this case being the terms of the continued coinciding for some number of terms

yes but I'm wondering about that type of statement for the continued fraction expansions

This would suggest what I said above is true, but we'd probably need uniform lower bounds on the number of terms required to get extra precision.

Give an irrational number, we can choose an interval small enough around it such that every point in that interval has the same decimal expansion up to some $n$th place right? There's a theorem (the name of which I can't recall right now), which says that each term of the continued fraction gives roughly ~1 decimal precision, and the worst case is generally with the golden ratio which for which we need something ~2.3 terms of the expansion to get an extra decimal digit of precision.

Hello! Is it true that for close enough, real, irrational numbers, their continued fractions are the same up to some $n$th element?

Yes I was looking at the wrong thing before, I can just use the usual length for my purposes. That being said, I'm curious if it's possible to define this non-overlapping curve rigorously. Intuitively you want a recursive definition, and ignore points you visited before. This would be an uncountably recursively define object, I'm not sure if there are things that could go wrong with that.

@TobiasKildetoft I think the reason why I wanted to do this in the first place is the wrong way to go about something, and I was making my life harder than I thought. Thanks for your thought anyway!!

OKay, so I think a simple way of saying what I need would be the length of the image of the curve, but I want it in terms of $\gamma ' (t)$.

No nevermind, this doesn't work. It only ignores the part of the curve the starts looping back on itself, but not the part of the curve that would start returning after moving back.

Ah maybe I can just define $X(t) = 0$ if $\gamma(t) < 0$, $X(t) = 1$ if $\gamma(t) \geq 0$ and then just use $\int_{0}^{1}|\gamma ' (t)|X(t)dt$.

I have a curve $\gamma\colon[0,1] \rightarrow \mathbb{R}$ that's $C^1$ but not necessarily simple. The length of this curve is $\int_{0}^{1}|\gamma^{'}(t)|dt$. But I want to measure the length of the curve which does not overlap with itself. It seems one can simply define a new curve ignoring the overlaps, which will still be at least piecewise $C^1$ and so we can measure it's length. Is there a name for this? Or a name for measuring the non-overlapping part of a curve?

Hey everyone!

I think I got it.

This is pretty simple but any idea how I can show that the iterates of points with non-zero imaginary part map to infinity under $f(z)=2z^2-1$? I can do it using some complex analysis as was suggested in one textbook, but it seems like an overkill. I can't get a elementary estimate right now though.

Hey guys!

@MatheinBoulomenos thank you for your proof regarding the nonexistence of representations. It's too high powered for me for sure, but I'll keep it in mind for the future.

@BalarkaSen I'm not sure if you saw my discussion with Mike yesterday, but to make things brief I was wondering if there are any nice representations of $Homeo^{+}(S^{1})$, the group of orientation preserving homeomorphisms of $S^1$.

Thanks! If you get any ideas/are curious about the details of the problem let me know! I'll ask my professor about this when I seem him tomorrow, he will probably have some ideas about this stuff.

I just figured having representation of this group would be a natural way of doing this, but since I don't know about rep theory beyond basic definitions, I wanted to ask if this seemed reasonable before I spent significant time learning parts of it.