Let $ X $ to be a compact metric space and let $ f: X rightarrow X $ to be an Anosov and transitive map.

Let $ x $ to be a generic point for that $$ mu_ {n_ {i}}: = frac {1} {n_ {i}} sum_ {j = 0} ^ {n_ {i} -1} delta_ {T ^ {j} (x) } rightarrow mu $$ for an invariant measure $ mu $.

It is well known that $ T $ satisfies the specification property. Let $ p in X $ be

a periodic point associated with $ delta, S $ and

$ {x, T (x) ,. . . , T ^ {n_ {i}} (x) } $

by the specification property. In particular, $ T ^ {n_ {i} + S} (p) = p $ and orbit of $ p $ coverage orbit of $ T ^ {n_ {i}} (x) $.

My first question is:

$ mu_ {n_ {i}, p}: = frac {1} {n_ {i} + S} sum_ {j = 0} ^ {n_ {i} + S-1} delta_ {T ^ { i} (p)} rightarrow mu $?

If yes. For $ mu in mathcal M (X) $ we write $ text {supp} mu $ for the *support from $ mu $*. It's easy to see that if a sequence of measurements $ ( mu_n) _ {n = 1} ^ infty $ low$ ^ * $ converges towards some $ mu in mathcal M (X) $then

$$

text {supp} , mu subset liminf_ {n to infty} ( text {supp} , mu_n).

$$

(By $ liminf $ above, I mean the lower limit of Kuratowski; for a definition, see entry in Wikipedia).

From what precedes, can we say $ supp mu $ is the periodical? Because $ mu_ {n_ {i}, p} $ has periodic support.