Consider the function f ( x ) : [ 0 , 1 ] → [ 0 , 1 ] given by { 2 x 0 ≤ x ≤ 1 2 x − 1 2 1 2 &lt; x ≤ 1 I found the measure with density given by ρ = 4 3 χ [ 0 , 1 2 ] + 2 3 χ ( 1 2 , 1 ] (with respect to Lebsegue measure) is invariant for this transformation. My question now is: how can I prove this system with this measure is ergodic? I thought to use the approach with invariant functions and Fourier series, but I'm not sure on how to write Fourier expansion with a measure different than Lebesgue's. I also thought to exploit a possible conjugacy with symbolic shift, but wasn't able to prove that [ 0 , 1 2 ] and ( 1 2 , 1 ] constitute a Markov partition of the unit interval. Any ideas?

Question

Consider the function   f  (  x  )  :  [  0  ,  1  ]  →  [  0  ,  1  ] given by      {                            2          x                          0          ≤          x          ≤                      1            2                                                x          −                      1            2                                                1            2                    &amp;lt;          x          ≤          1                        I found the measure with density given by   ρ  =      4    3        χ          [      0      ,              1        2            ]        +      2    3        χ          (              1        2            ,      1      ]       (with respect to Lebsegue measure) is invariant for this transformation. My question now is: how can I prove this system with this measure is ergodic? I thought to use the approach with invariant functions and Fourier series, but I&#039;m not sure on how to write Fourier expansion with a measure different than Lebesgue&#039;s. I also thought to exploit a possible conjugacy with symbolic shift, but wasn&#039;t able to prove that   [  0  ,      1    2    ] and   (      1    2    ,  1  ] constitute a Markov partition of the unit interval. Any ideas?

Bruno Dixon · Accepted Answer

They gave as the exact same question in the course &quot;introduction to analysis&quot;, as an example for uses for Fourier series, so the solution is bases on Fourier series.Let&#039;s take   f  :      R    →      C   periodic and continuous such that       ∫          0              2      π            |    f  (  x  )  −  f  ∘  T  (  x  )            |        2    =  0, we want to show that there exists   c  ∈      C   such that   f  ≡  c.From Parseval&#039;s identity, we know that       ∫          0              2      π            |    f  (  x  )  −  f  ∘  T  (  x  )            |        2    =      ∑          k      ∈      Z            |                      f        −        f        ∘        T            ^        (  k  )            |        2      .Therefore we can infer that                     f        −        f        ∘        T            ^        (  k  )  =  0  ⇒  (  b  y     l  i  n  e  a  r  i  t  y  )               f      ^        (  k  )  =                    f        ∘        T            ^        (  k  )     ∀  k  ∈      Z  After doing some calculation we can also infer that for any &quot;even&quot;   k  ∈      Z   ,                     f        ∘        T            ^        (  k  )  =            f      ^        (      k    2    ).so by induction we can infer now that for any   p  ≠  0,                     f        ∘        T            ^        (      2    n    ⋅  p  )  =            f      ^        (  p  ) From Riemann–Lebesgue lemma, we can now infer that      lim          n      →      ∞                          f        ∘        T            ^        (      2    n    ⋅  p  )  =  0  ⇒            f      ^        (  p  )  =  0  ,     ∀  p  ≠  0So now we conclude that Fouriee series of f is             f      ^        (  0  ) and therefore   f  ≡            f      ^        (  0  ), so f is constant, as required.

Consider the function f ( x ) : [ 0 , 1 ] → [

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