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For each pseudo‐Anosov map    ϕ  :  S  →  S   , we associate it with a   Q  ‐vector space lying in   R  , and denote it by    A  (  S  ,  ϕ  )   . The invariant    A  (  S  ,  ϕ  )    is defined by an interaction between the Thurston norm and the dilatation of pseudo‐Anosov maps. We develop a few nice properties of    A  (  S  ,  ϕ  )    and give a few examples to show that    A  (  S  ,  ϕ  )    is a nontrivial invariant. These nontrivial examples give an answer to a question asked by McMullen, and show that the minimal point of the restriction of the dilatation function on the fibered face need not be a rational point.

A transcendental invariant of pseudo‐Anosov maps