# The morphology of MSS-sequences in a wide class of unimodal maps, its structure and decomposition

## TL;DR

This paper analyzes the structure and decomposition of MSS-sequences in unimodal maps, providing explicit forms, theorems for their decomposition, and algorithms for constructing these sequences.

## Contribution

It introduces a detailed explicit structure of MSS-sequences, including decomposition theorems and an algorithm for their construction, advancing understanding of their combinatorial properties.

## Key findings

- Explicit structure of MSS-sequences provided
- Decomposition theorems for non-primary MSS-sequences
- Algorithm for constructing sequence blocks

## Abstract

The MSS-sequences (U-sequences) in a wide class of unimodal maps have the look $\mathrm{P}=(\mathrm{R} \mathrm{L}^{q})^{n_1} \mathrm{S}_1(m_1,q-1) (\mathrm{R} \mathrm{L}^{q})^{n_2}\mathrm{S}_2(m_2,q-1) $ $\ldots$ $ (\mathrm{R} \mathrm{L}^{q})^{n_r}$ $ \mathrm{S}_r(m_r,q-1)\mathrm{C},$ where $\mathrm{S}_i(m_i, q-1)$ are sequences of $\mathrm{R}$s and $\mathrm{L}$s that contain at most $q-1$ consecutive $\mathrm{L}$s. The first block $\mathrm{RL}^q$ and the sequence $\mathrm{S}_1$ following it are essential for an admissible sequence to be a MSS-sequence. Moreover $\mathrm{S}_i(m_i,q-1), \ i=2, \ldots, r$ are determined by $\mathrm{S}_1(m_1,q-1)$. Explicit structure of MSS-sequences will be given as well as the theorems that decompose the non-primary MSS-sequences. The cardinality will be calculated for some important sets of non-primary MSS-sequences and an algorithm to generate the blocks $\mathrm{S}_i(m_i,q-1), \ i=1, \ldots, r$ will be provided, as the construction of the blocks $\mathrm{S}_i(m_i,q-1)$ allows the construction of the MSS-sequences.