Bitcoin can be represented on the Ethereum network using several methods. The easiest way to do it is to implement a 2 way peg - where coins on the Bitcoin blockchain are frozen into a address, then generated as tokens on the Ethereum blockchain. If you want to then access the original Bitcoins, you can provably burn/destroy the Ethereum-Bitcoin tokens unlocking the originals and allowing them to be transacted again.
Alternatively, you can do it by taking a snapshot of the Bitcoin blockchain and allowing holders to redeem Bitcoin tokens based on the wallet information inside the Ethereum ecosystem. This method however is limited as it does not allow for any changes in the Bitcoin blockchain, and the price of said snapshot-coins will be independent of those on the Bitcoin blockchain.
Most importantly though, both methods allow Bitcoin to interact with a smart contract capable blockchain, enriching it as opposed to directly competing with it.
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Alternatively, you can do it by taking a snapshot of the Bitcoin blockchain and allowing holders to redeem Bitcoin tokens based on the wallet information inside the Ethereum ecosystem. This method however is limited as it does not allow for any changes in the Bitcoin blockchain, and the price of said snapshot-coins will be independent of those on the Bitcoin blockchain.
Most importantly though, both methods allow Bitcoin to interact with a smart contract capable blockchain, enriching it as opposed to directly competing with it.