The Set of Elements That Do Not Escape the Tent Map Is a Cantor Set
I have a relatively simple proof for all $r>2$. There is no need for $r$ to be odd, nor integer, rational, etc. We only need $r$ to be real and greater than two. I think Epsilon's answer confused $s$ with $r$, since he thought $fracr-12rin C;(wrong)$, which it does not , the point he describes is $frac1rin C$. As stated in the question, the set of $x$ with bounded orbits is in fact a Cantor set with $r=frac2ss-1$, but, in order to avoid the confusion of the previous post, I will provide the definition of the Cantor set in terms of $r$. First, $C_0 =,$. Then, $$forall nin mathbbZ_:C_n=left(frac1rC_n-1
ight)cupleft(fracr-1rfrac1rC_n-1
ight);.;(1)$$ You can check that this definition coincides with the one of the question if $r=frac2ss-1$. We need $r>2$ so that the Cantor set just defined is a proper subset of the $,$ interval. In fact:The sketch of the proof is as follows:
1. Understanding concatenating the empty set to any set.
It turns out from the comments that the context is regular sets. If $A$ and $B$ are sets, we define $Acirc B=ab:ain Atext and bin B$. If $B=varnothing$, there are no objects $bin B$, so there are no objects $ab$ such that $ain A$ and $bin B$; thus, $Acircvarnothing=varnothing$.
2. Write an open set in terms of a closed set
$A$ is closed if $A^c$ is open. Remember that in a metric space, a set is open if for each point $y$ there is an $epsilon$ such that the $epsilon$-ball around $y$ lies completely in the set.So let $y$ be any point in $A^c$. There is an $n_0 in mathbbN$ such that $d(y,a) geq r frac1n_0$. For any point $x in A$ it holds that $d(x,a) r frac1n_0 - r - frac12 n_0 = epsilon$$ follows. Therefore, the $epsilon$-ball $K_epsilon := y' in X: d(y',y)
3. A set containing one element is an open set. Why?
In general, it is not the case that single-element sets are open. For example, in the usual topology on $Bbb R$, the one induced by the usual metric, single-element sets are not open; open sets are unions of open intervals, and every open set (except $varnothing$) is infinite.Say we have a metric space with the so-called "discrete" metric $d$. Recall that this means that $$d(x,y) = begincases 0, textif $x = y$ 1, textif $x
e y$endcases$$ A metric space has a natural topology "induced by" its metric. The "metric topology" induced by the metric $d$ is the one that has as its basis all "balls" $N_d,epsilon(x)$ where $$N_d,epsilon(x) = p mid d(x,p)
4. Generate a set of combinations
I highly recommend that you make two public functions, that return different data. I would also not name a public function genCombinations2, instead use combinations.You should make the combinations function work the way your code does when elemsSize = undefined. As handeling the elemsSize should be a different problem.And so you would have:From this, you want to group your values. This is two loops over your data:And so you just need to think of a way to create combinations. One way could be to do the same as Python's itertools.combinations: (Note the following is not tested.)
