In this talk, I will present the main questions arising in the dynamics (iterations) of transcendental maps in ℂ. After introducing the basic concepts and examples, I will explain the main ideas on the research project which was originated by our joint paper with Mariusz Urbański more than ten years ago. The approach proposed in our first paper on this project turned out to be fruitful and stimulating and our work was followed by a series of further results, which, gradually, contribute to the theory of measureable dynamics of transcendental maps. I will sketch the results obtained lately in my joint works with K. Barański and B. Karpińska. In particular, I will discuss the concept of conformal measure in the setting of the dynamics of transcendental maps. Some open questions will be also discussed. The talk will be accessible to non-specialists.
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