Yes, exactly same events repeat.
Also discussed in Does "Free Will" really exist, even if universe repeats itself in perfect pattern? and other related Qn-s.
BG 8.17 - Knowing that thousand eras constitute a day of Brahman, [and] thousand eras complete a night, are the people who know day, [and] night.
BG 8.18 - On arrival of day, all manifestations originate from "Unmanifest"; On arrival of night they annihilate into [what is] known as "Unmanifest" only.
BG 8.19 - This world of beings only happens again & again; Annihilates upon arrival of night, [and] originates upon arrival of day.
BG 9.10 - Under "My" supervision, the Prakruti produces moving & non-moving [beings]; Due to this reason the world revolves
That means, whatever Shri Rama, Krishna, Shankara, Ganesha did during their live, exactly same things they will do in the next day of Brahma as well.
Not only them, you will be asking this same Qn and I will be writing this same answer.
This philosophy of Eternal Return is also known as "kAla chakra" by few.
On the rational side you may think like this:
- The time is flowing continuously; Infinite time has passed before this moment and infinite time will pass after this moment
- We are present in the current moment. How is that possible, if there is infinite time to be passed before current moment?
- That means, there has to be a "timespan" (i.e. collection of finite events), which is repeating infinitely
- With point 3, the time also becomes infinite (timespan x infinite) and we can also be present as current moment in the timespan
- This "timespan" can be called the day of Brahma; The night of Brahma will be equally long, but nothingness pervades during that duration
That's why Shri Krishna told to Arjuna in the beginning of Gita:
BG 2.12 - But certainly never was a time, when I did not exist, nor you, nor these rulers of men. And surely it is not that we all shall cease to exist after this.
Which means that all had existed in past cycles and will continue to exist in future cycles as well.