# invariant measures of the expanding maps on the circle

I would be very happy to know about original references for the following results;

For the expanding map $x \mapsto mx$ on the circle, (with $m$ some integer greater than 1)

(1) There exist uncountably many ergodic invariant probability measures.

(2) Atomic invariant measures are weak star dense in the set of all invariant probability measures.

Thank you very much!

• To the best of my knowledge (2) has not been published at all per se. The proof of this result is more-or-less identical to that for the shift transformation, which is due to Parthasarathy (On the category of ergodic measures, Illinois J. Math. 5 (1961) 648–656). – Ian Morris Dec 19 '14 at 12:43
• The existence of uncountably many ergodic invariant probability measures might also be difficult to find a crisp reference for. This result is true for the following reason. Let us specialise to the doubling map. Give the dyadic interval $(k/2^n,(k+1)/2^n)$ measure $p^i(1-p)^{n-i}$ where $i$ is the number of ones in the binary expression of $k$. Then this extends to a doubling-map-invariant, mixing (hence ergodic) probability measure on the circle, for every fixed $p \in (0,1)$. Obviously these measures are distinct for different $p$. – Ian Morris Dec 19 '14 at 16:45
• Thank you so much for the reference and the example. It has been a hard problem to find a reference. – user20471 Dec 20 '14 at 2:35

## 1 Answer

The proof of (2) is contained in the much more general theorem due to Sigmund. This is because the main result of Sigmund "Generic Properties Of Invariant Measures for Axiom A-Diffeomorphisms" Inventiones Math. 11 (1970), pp. 99-109 applies. One may argue that Sigmund considers only toral automorphisms, but his reasoning is based on the fact that the maps he considers have the periodic specification property. The proof that periodic specification sufficies for (2) to hold can be found in Ergodic theory on compact spaces (Volume 527 of Lecture notes in mathematics) by Manfred Denker, Christian Grillenberger, Karl Sigmund (Springer-Verlag, 1976). It is easy to see that maps $x\mapsto mx$ ($m>1$) have the periodic specification property (a theorem of Blokh stating that topological mixing implies specification for maps on graphs can also be invoked, and to a proof that $x\mapsto mx$ have topological mixing is an exercise). In any case, much more is known and Ian Morris above is right - the idea follows Parthasaraty's article.

P.S. and (1) follows from another Sigmund paper ON THE CONNECTEDNESS OF ERGODIC SYSTEMS or can be deduced form the general result: if a nontrivial simplex of invariant measures (known to be a Choquet simplex see inverse problem for ergodic measures) has a dense set of ergodic measures (extreme points), then the set of extreme points must be arcwise connected and hence uncountable. This is a consequence of the Lindenstrauss, Olsen and Sternfeld result (The Poulsen simplex, Annales de l'institut Fourier 28.1 (1978): pp. 91-114).