How can i have a one-day sale on Amazon?

[Mellie]

You apply for an ad spot and if successful, you discount the price of the book on your KDP and B&N dashboard at least 12 hours before the ad runs.

I don't even know what you mean by "price cycling" but if you're looking at the book page in different Amazon stores I suspect you are simply seeing the effect of foreign exchange rates fluctuating.

[Mellie]

If you are with a publisher then it is up to them to organise and discount price accordingly. I'm surprised they are not marketing and applying for advertising spots for their catalogue (which is what other publishers do).

Are you talking paperback price? Amazon applies a discount to the retail price based on its own algorithms. No, it does not discount in response to demand. I'm not aware of any service that will email you about price. If you want to offer your paperback to readers at a discount, why don't you buy in bulk from your publisher (mine used to offer an author discount) and then sell through your website?

[Debbie's Spurts (D.A.)]

Amazon does price match so sometimes a price drop is because book was cheaper or discounted elsewhere--no clue if that's why it happened with your particular book.