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It is well known that  C  2 -transformation  φ  of the unit interval into itself with a Markov partition  (2.1)   π  = { I k   :  k  ∈  K } admits  φ -invariant density  g  ( g  ≥ 0, ∥ g ∥ = 1) if:  (2.2)  ∣( φ n  )′∣ ≥  C  1  &gt; 1 for some  n  ( expanding condition );  (2.3)  ∣ φ ″( x )/( φ ′( y )) 2 ∣ ≤  C  2  &lt; ∞ ( second derivative condition ); and  (2.4)  # π  &lt; ∞ or  φ  ( I k  ) = [0, 1], for each  I k   ∈  π . If  (2.4)  is deleted, then the situation dramatically changes. The cause of this fact was elucidated in connection with so-called  Adler’s Theorem  ([ 1 ] and [ 2 ]).

A few problems connected with invariant measures of Markov maps - verification of some claims and opinions that circulate in the literature