Does the map() function works on an ATtiny85? I have a working project on Arduino Uno.Now I want to shrink my project using an ATtiny85. Before purchasing an waiting for the shipment to arrive, I want to make sure that my whole code will run on ATtiny85.

specialy, will the map() function which is working on  an Arduino Uno ( ATMEGA328P-PU ),  work with a ATtiny85 Microcontroller?
The map function uses integer arithmetic, and since the ATmega series and the ATtiny series use the same 8-bit AVR instruction set, there should be no difference when using it, and you are good to go from that aspect.

Consider that there are many other differences regarding I/O pins, storage etc. - that you might want to consider. Here is a good source for a basic comparison between the ATmega based Arduino to ATtiny85.