Absolutely continuous invariant measures for a class of affine interval exchange maps

Abstract

We consider a class <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of affine interval exchange maps of the interval and we analyse several ergodic properties of the elements of this class, among them the existence of absolutely continuous invariant probability measures. The maps of this class are parametrised by two values <italic>a</italic> and <italic>b</italic> , where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a comma b element-of left-parenthesis 0 comma 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">a,b \in (0,1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . There is a renormalization map <italic>T</italic> defined from <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to itself producing an attractor given by the set <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper R"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">R</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {R}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of pure rotations, i.e. the set of ( <italic>a, b</italic> ) such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="b equals 1 minus a"> <mml:semantics> <mml:mrow> <mml:mi>b</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo> − </mml:mo> <mml:mi>a</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">b = 1 - a</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The density of the absolutely continuous invariant probability and the rotation number of the elements of the class <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are explicitly calculated. We also show how the continued fraction expansion of this rotation number can be obtained from the renormalization map.