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PREPRINTS IN THIS SERIES, IN PDF FORMAT.
* Starred papers have appeared in the journal cited.
We introduce puncture-forgetting maps for measured foliations and investigate their relations with mapping class groups, Teichmüller spaces, and extremal length. To this end, we develop the notions of cube complexes of pre-homotopic multicurves and tree coordinate systems on CAT(0) cube complexes. As an application to the dynamics of post-critically finite rational maps on the Riemann sphere, we o… ▽ More
Submitted 22 September, 2026; v1 submitted 29 July, 2026; originally announced July 2026.
Comments: 71 pages, 15 figures
MSC Class: 30F60; 57K20; 37F10
Given a quadratic polynomial with an irrationally indifferent fixed point of bounded type, a bubble is an iterated preimage of its Siegel disk. Similar to pseudo-Siegel disks, one constructs pseudo-bubbles from bubbles by filling in parabolic fjords. We introduce the near-degenerate regime for the α′-Dynamics controlling the geometry of these pseudo-bubbles in deep scale. Consequently, we obtain uniform bounds on the size and regularity of pseudo-bubbles independent of the rotation number. In [DLL], jointly with Lyubich, this result serves as a necessary ingredient for establishing uniform butterfly bounds and ultimately the combinatorial rigidity of the full attractor of neutral renormalization.▽ More
Submitted 14 September, 2026; originally announced September 2026.
Comments: 33 pages, 11 figures
MSC Class: 37F50; 37F25; 37F31; 37F10
We prove combinatorial rigidity for the full renormalization attractor of neutral quadratic polynomials. More precisely, any two bi-infinite renormalization towers with the same combinatorics are conformally conjugate on neighborhoods of their Mother Hedgehogs. The rigidity theorem includes arbitrary irrational combinatorics and respective parabolic enrichments. The proof relies on a comprehensive analysis of neutral cascades, i.e. transcendental dynamical systems arising from the rescaled limits of the first return maps of neutral quadratic polynomials. We establish uniform butterfly bounds for neutral cascades and prove that any two combinatorially equivalent neutral cascades are affinely conjugate. We also prove rigidity for parabolic towers with equivalent backward combinatorics. △ Less
Submitted 29 September, 2026; originally announced September 2026.
Comments: 118 pages, 23 figures
MSC Class: 37F25; 37E20; 37F50; 37F40; 37F10
We study the behavior of currents on flag-manifolds under actions of Anosov subgroups of complex semisimple Lie groups G. Given a current T of bidimension (k, k) on the full flag-manifold F = G/B, we average it under the group action via a construction analogous to the construction of Patterson-Sullivan measures. We show that under certain genericity conditions (in the case of currents of integration over subvarieties), the limiting current is a Gibbs current, i.e. is given by integration (with respect to a Gibbs measure) over the flag-limit set of suitable multiples of currents of integration along Schubert varieties based at the limit points of the group. We prove that the same equidistribution result (but without any genericity assumptions) for currents defined by smooth forms on F. We also prove a form of Axiom A property for the suspension flow of the group action on F.
Submitted 24 July, 2026; originally announced July 2026.
MSC Class: 37C85; 37D40; 32M05
We prove 
bounds and MLC (local connectivity of the Mandelbrot set 
) for a class of infinitely renormalizable parameters whose renormalization type is primitive but can approach the cusp of . To this end we develop and refine a variety of tools that allow us to control degeneration of renormalizations. They include the Thin-Thick Decomposition, the Value Calculus, the Wanderers Theorem, and the Wave Lemma. △ Less
Submitted 25 June, 2026; originally announced June 2026.
In this paper we prove a priori bounds for an ``elephant eye'' combinatorics. Little 
-copies specifying these combinatorics are allowed to converge to the cusp of the Mandelbrot set. To handle it, we develope a new geometric tool: uniform thin-thick decompositions for bordered Riemann surfaces.
Submitted 29 January, 2026; originally announced January 2026.
In this Note, we present recent developments in the Renormalization Theory of quadratic polynomials and discuss their applications, with an emphasis on the MLC conjecture, the problem of local connectivity of the Mandelbrot set, and on its geometric counterparts.
Submitted 30 December, 2025; originally announced December 2025.
We extend uniform pseudo-Siegel bounds for neutral quadratic polynomials to 
-quadratic-like Siegel maps. In this form, the bounds are compatible with the -quadratic-like renormalization theory and are easily transferable to various families of rational maps. The main theorem states that the degeneration of a Siegel disk is equidistributed among combinatorial intervals. This provides a precise description of how the -quadratic-like structure degenerates around the Siegel disk on all geometric scales except on the ``transitional scales'' between two specific combinatorial levels.
Submitted 30 September, 2025; v1 submitted 26 September, 2025; originally announced September 2025.
We present a proof of the conjecture by Bonifant and Milnor (see arXiv:2503.08868) regarding the similarity between the connectedness locus of the curve 
at Misiurewicz parameters and their corresponding filled Julia sets in a neighborhood of the corresponding free co-critical point. The proof is in parallel with the generalization of Tan Lei's proof of similarity in the Mandelbrot set developed by Kawahira.
Submitted 30 October, 2025; originally announced October 2025.
We establish certain uniform a priori bounds for hyperbolic components of disjoint type. As an application, we will prove that Sierpinski carpet hyperbolic components of disjoint type are bounded. Furthermore, we show that for each map 
on the closure of such a hyperbolic component, there exists a quadratic-like restriction around every non-repelling periodic point. Extensions of these results to non-Sierpinski configurations are underway. As a prototype example, we describe the post-critical set of any map on the boundary of the hyperbolic component of 
.
Submitted 29 September, 2025; originally announced September 2025.