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YCor
YCor
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This$\DeclareMathOperator{\Inv}{Inv}\DeclareMathOperator{\Erg}{Erg}$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 $\DeclareMathOperator{\Inv}{Inv}$ $\DeclareMathOperator{\Erg}{Erg}$ 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 $[0,1]$?

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 $\DeclareMathOperator{\Inv}{Inv}$ $\DeclareMathOperator{\Erg}{Erg}$ 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 $[0,1]$?

$\DeclareMathOperator{\Inv}{Inv}\DeclareMathOperator{\Erg}{Erg}$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 $[0,1]$?

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Liviu Nicolaescu
Liviu Nicolaescu
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Ergodic measures for the logistic map

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 $\DeclareMathOperator{\Inv}{Inv}$ $\DeclareMathOperator{\Erg}{Erg}$ 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 $[0,1]$?