Abstract: The Near-Degenerate Regime, i.e., a collection of tools and ideas to control degenerations of Riemann surfaces endowed with dynamics, has been instrumental in the most recent progress towards the MLC Conjecture ("the Mandelbrot set is locally connected") and understanding rigidity of quadratic polynomials. Likewise, it was one of the key ingredients of extending Yoccoz’s results from quadratic polynomials to polynomials of higher degree. In joint work in progress with Dima Dudko, we introduce this set of ideas into Transcendental Dynamics, specifically focusing on a family of Yoccoz maps with bounded criticality in the Eremenko–Lyubich class. This approach allows us to construct dynamically meaningful polynomial approximations of transcendental entire maps within our class, thereby elevating and extending the most fundamental polynomial results to the transcendental setting, for the first time in a genuinely complex Yoccoz situation (no neutral or infinitely renormalizable dynamics, but otherwise, no restrictions on the postcritical set). Among the consequences are the realization of all combinatorics, rigidity, and the density of hyperbolicity (near Yoccoz parameters). For postcritically finite maps, our results yield a transcendental version of W. Thurston’s Characterization Theorem, thus extending the classical finite-degree setting to branched coverings of infinite degree. In the talk, I will outline this circle of results.

Zoom Meeting ID: 181 155 584