# plotting complicated function

I would like to approximate the sum $$h(a,x) = \frac{-2}{n} \sum_{n=0}^{n-1} log|T_a^n(x)|$$ where $n$ is large like $n= 1000 - 5000$ and for a fixed $a$ $$T_a(x) = \Big|\frac{1}{x}\Big| - \Big\lfloor{\Big|\frac{1}{x}\Big| - 1 +a}\Big\rfloor$$ where $x \in (0,1).$

By fixing $x$ to be a value $x_0 \in (0,1)$, e.g. $x_0 = 1/\pi$, $$h(a, x_0) = h(a, 1/\pi)$$ a function of one variable, and I want to plot a 2D graph of point $(a, h(a, 1/\pi))$, by fixing $n = 2000$, for $a \in [0,1].$

I figure how to calculate the value at one given $a$ using SageMath, for example, when $a = 1$,

T(x) = 1/x - floor(1/x)
s = 0
for k in xrange(0, 1000):
a = 0
a = nest(T, k, 0.79)
b = abs(a)
c = log(b)
s = s + c


Then $\frac{-2}{1000}s$ give the approximation for the sum when $x = 0.79$, $n = 1000$, $a = 1$.

But for plotting, I think I need to define the function $h(a, x)$ which is a summation over composition of functions. I tried to use sum and symbolic_sum but failed.

Any help how to achieve this please?

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I suspect that you want to write $|T_a(x)|^k$ instead of $T_a^n(x)$, right? Otherwsie, what means $T^n$ here?
@Masacroso: it means the function $T_a$ is applied $n$ times. For example, $T_a^2(x) = T_a(T_a(x))$.
well, in first place note that $$T_a(x)=x^{-1}-\lfloor \lfloor x^{-1}\rfloor +\{x^{-1}\}-r\rfloor=\{x^{-1}\}-\lfloor \{x^{-1}\}-r\rfloor=\{x^{-1}\}+[r>\{x^{-1}\}]$$ for $r\in[0,1]$ and where $[\cdot ]$ is an Iverson bracket and $\{x^{-1}\}$ is the fractional part of $x^{-1}$. But then note that $T_a(T_a((x))$ is not defined when $\{ x^{-1} \}=0$ and $r\le \{x^{-1}\}$.