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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.)

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