$\DeclareMathOperator{\Inv}{Inv}\DeclareMathOperator{\Erg}{Erg}$This is mostly curiosity on my part and I hope that the MO community might be able to help.

For $c\in (0,4]$ consider the logistic map $$ T_c:[0,1]\to[0,1],\;\;T_c(x)=cx(1-x). $$ Denote by $\Inv_c$ the collection of Borel probability measures on $[0,1]$ that are $T_c$-invariant and by $\Erg_c\subset \Inv_c$ the subset of consisting of ergodic ones.

**Question 1.** There are simple examples such ergodic measure
measure, e.g., measures concentrated on a periodic orbit. Do there exist ergodic measures not concentrated on a periodic orbit?

**Question 2.** This is a bit more vague. Is it known how the set $\Erg_c$ evolves with changing $c $ inside the set Borel probability measures on $[0,1]$?