Subsystems of transitive subshifts with linear complexity

We bound the number of distinct minimal subsystems of a given transitive subshift of linear complexity, continuing work of Ormes and Pavlov [On the complexity function for sequences which are not uniformly recurrent. Dynamical Systems and Random Processes (Contemporary Mathematics, 736). American Mathematical Society, Providence, RI, 2019, pp. 125--137]. We also bound the number of generic measures such a subshift can support based on its complexity function. Our measure-theoretic bounds generalize those of Boshernitzan [A unique ergodicity of minimal symbolic flows with linear block growth. J. Anal. Math.44(1) (1984), 77–96] and are closely related to those of Cyr and Kra [Counting generic measures for a subshift of linear growth. J. Eur. Math. Soc.21(2) (2019), 355–380].

Stopping Transformed Growth with Cytoskeletal Proteins: Turning a Devil into an Angel

SummaryThe major hallmark of cancer cells is uncontrollable growth on soft matrices (transformed growth), which indicates that they have lost the ability to properly sense the rigidity of their surroundings. Recent studies of fibroblasts show that local contractions by cytoskeletal rigidity sensor units block growth on soft surfaces and their depletion causes transformed growth. The contractile system involves many cytoskeletal proteins that must be correctly assembled for proper rigidity sensing. We tested the hypothesis that cancer cells lack rigidity sensing due to their inability to assemble contractile units because of altered cytoskeletal protein levels. In four widely different cancers, there were over ten-fold fewer rigidity-sensing contractions compared with normal fibroblasts. Restoring normal levels of cytoskeletal proteins restored rigidity sensing and rigidity-dependent growth in transformed cells. Most commonly, this involved restoring balanced levels of the tropomyosins 2.1 (often depleted by miR-21) and 3 (often overexpressed). Restored cells could be transformed again by depleting other cytoskeletal proteins including myosin IIA. Thus, the depletion of rigidity sensing modules enables growth on soft surfaces and many different perturbations of cytoskeletal proteins can disrupt rigidity sensing thereby causing transformed growth of cancer cells.

Infinite interval exchange transformations from shifts

We show that minimal shifts with zero topological entropy are topologically conjugate to interval exchange transformations, which are generally infinite. When these shifts have linear factor complexity (linear block growth), the conjugate interval exchanges are proved to satisfy strong finiteness properties.

On the number of ergodic measures for minimal shifts with eventually constant complexity growth

In 1985, Boshernitzan showed that a minimal (sub)shift satisfying a linear block growth condition must have a bounded number of ergodic probability measures. Recently, this bound was shown to be sharp through examples constructed by Cyr and Kra. In this paper, we show that under the stronger assumption of eventually constant growth, an improved bound exists. To this end, we introduce special Rauzy graphs. Variants of the well-known Rauzy graphs from symbolic dynamics, these graphs provide an explicit description of how a Rauzy graph for words of length $n$ relates to the one for words of length $n+1$ for each $n=1,2,3,\ldots \,$.