# Bitcoin's dominance in the context of an ever-expanding number of cryptocurrencies

To express Bitcoin's dominance in the context of an ever-expanding number of cryptocurrencies using mathematical notions of infinity and asymptotic behavior, let's construct a model where the denominator, representing the total market capitalization of all cryptocurrencies, is viewed as a floating, dynamically changing value. This approach highlights the notion that Bitcoin's dominance is in a continuous state of discovery against an unknown and increasing number of present and future cryptocurrencies.

### Mathematical Formulation

1. **Dynamic Denominator as an Asymptotic Series**:
Consider the total market capitalization of all cryptocurrencies, \( M(t) \), as a function of time that incorporates both existing and forthcoming cryptocurrencies. This can be expressed as:
$$
M(t) = \sum_{i=1}^{N(t)} C_i(t)
$$
where \( N(t) \) is a function representing the count of cryptocurrencies at time \( t \), and \( C_i(t) \) is the market cap of the \( i \)-th cryptocurrency. As \( t \) progresses, \( N(t) \) approaches infinity, symbolizing the continuous introduction of new cryptocurrencies.

2. **Floating Denominator and Bitcoin Dominance**:
The dominance of Bitcoin, \( D(t) \), then can be defined as:
$$
D(t) = \frac{B(t)}{M(t)}
$$
where \( B(t) \) is the market cap of Bitcoin. Given \( M(t) \)'s continuous growth as \( N(t) \) increases, \( D(t) \) must be continuously recalculated against this evolving backdrop.

3. **Asymptotic Behavior Analysis**:
The concept of asymptotic analysis becomes crucial when considering how \( D(t) \) behaves as \( N(t) \) increases without bound. One might consider:
$$
\lim_{N(t) \to \infty} D(t) = \lim_{N(t) \to \infty} \frac{B(t)}{\sum_{i=1}^{N(t)} C_i(t)}
$$
This limit will help us understand the long-term behavior of Bitcoin's dominance as the market expands potentially indefinitely.

### Implications for Investment and Market Analysis

- **Investment Universe in Discovery Mode**: As the denominator \( M(t) \) is not fixed but increases with \( N(t) \), Bitcoin's dominance metric is continuously in discovery mode. This reflects a market in flux where Bitcoin is competing against an ever-increasing and unknown number of cryptocurrencies.

- **Necessity for Dynamic Models**: Traditional static models of market dominance are inadequate in this context. Instead, dynamic models that can update in real-time and accommodate an expanding universe of assets are necessary to provide accurate and relevant metrics.

- **Strategic Adjustments**: For investors and market analysts, understanding that Bitcoin's dominance is not a static figure but one that fluctuates against a backdrop of continual market expansion can guide more adaptive and forward-looking investment strategies.

In summary, considering Bitcoin's dominance from this perspective underscores the fluidity of the cryptocurrency market and highlights the need for investment strategies and analytical models that can dynamically adjust to an ever-expanding investment universe. This approach leverages mathematical concepts of infinity and asymptotic behaviors to capture the ongoing competitive landscape of Bitcoin against current and yet-to-emerge cryptocurrencies.