Invariant densities for composed piecewise fractional linear maps
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Invariant densities for composed piecewise fractional linear maps Fritz Schweiger1 Received: 22 February 2019 / Accepted: 3 November 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract The starting point for this paper was the following problem: If we know the invariant densities for two maps S and T can we say something about the invariant density of the map U = T ◦ S? This problem seems not to be touched on within ergodic theory. In this note we look at the inverse problem. Let the invariant density for U = T ◦ S be known. What can we say about invariant densities for S and T ? We discuss a simple model, namely the class of all piecewise fractional linear maps with two branches on the unit interval [0, 1]. Keywords F-expansion · Piecewise fractional linear maps · Invariant measures Mathematics Subject Classification 11K55 · 28D05 · 37A05
0. Introduction The map T x = x1 − a, a = a(x) = x1 generates the continued fraction algorithm. 1 Its invariant density is well known: h(x) = 1+x . A natural generalization to dimension 2 is the Jacobi algorithm [4]. It can be generated by the map T (x, y) =
1 y 1 y − a, − b , a = a(x, y) = , b = b(x, y) = . x x x x
This map admits an invariant density but no explicit expression is known for it. It is a surprise that the map T 2 can be written as a composition of two maps with well known densities. Since T 2 is ergodic the finite invariant measure for T 2 is also invariant for T . Let T1 be the skew product
Communicated by H. Bruin.
B 1
Fritz Schweiger [email protected] FB Mathematik, University of Salzburg, Hellbrunnerstr. 34, 5020 Salzburg, Austria
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F. Schweiger
T1 (x, y) =
1 y y 1 − b, − a , a = a(x, y) = , b = b(x, y) = . x x x x
We note that a(x, y) = b(x, y) implies a(T1 (x, y)) = 0. The inverse branches of T1 are given by the matrices ⎛
b V1 (a, b) = ⎝ 1 a
⎞ 0 0⎠ 1
1 0 0
(see the remarks on fibred systems given below and the references mentioned there). The invariant measure for T1 has the following density. 1 1 + ,x
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