$z_{n+1} = v z_n ^5 - (v-1) z_n $ with $ v = 3,06328648997749...$

Question

Consider the set of irrational numbers between $-1$ and $1$ and call it $T$.

Let $4>v>1$ and consider the iterations

$ x_0$ belongs to $T$. $$ z_0 = x_0 $$ $$z_{n+1} = v z_n ^5 - (v-1) z_n $$

Let $[a,b]$ be a non-empty open interval where $-1 < a < b < 1$.

There exists a largest possible value $t$ - probably Unique - for $v$ such that

For almost all values $x_0$ and any values $a,b$ :

$z_n$ is dense in $[a,b]$.

It appears that this largest value for $v$ is equal to

$$ t = 3.06328648997749... $$

I assume there is no closed form for $z_n$ with $ v = t $.

But what is the closed form for this mysterious $ t $ ?

I considered algebraic forms with fifth roots. I thought about derivatives. I even considered a connection to the feigenbaum constants.

Is $t$ algebraic or transcendental ?

Is it related to areas of some fractals and julia sets ?

Not only do i lack a closed form , I also have no efficient method to compute $t$.

Is it related to a koenigs type lim or function ?

Is there an expression from calculus like an infinite sum ?

Is this related to The paris constant ?

I read about chaos theory , oscillations , dynamical systems , fractals , control theory and optimization. But without succes.

Maybe this relates to sums of 5 exponentials ?

Also i wonder about its continued fraction. Is it normal in the khinchin sense ?