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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 try to use sum and symbolic_sum but fail.

Any help how to achieve this please ?

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, SageMath, for example, when $a = 1$,

T(x) = 1/x - floor(1/x)
 
s=0
 
floor(1/x)
s = 0
for k in xrange(0,1000):
 
 a=0

 a=nest(T, xrange(0, 1000):
     a = 0
     a = nest(T, k, 0.79)

 b=abs(a)

 c=log(b)

 s=s+c
     b = abs(a)
     c = log(b)
     s = s + c
 
Then $\frac{-2}{1000}s$ give the approximation for the sum when $x = 0.79, n =1000, a=1$.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 try tried to use sum sum and symbolic_sum symbolic_sum but fail.failed.
 
Any help how to achieve this please ?please?