The topological entropy of uniformly continuous selfmaps of uniform spaces is defined following Bowen-Dinaburg approach. It works particularly well for continuous endomorphisms of locally compact groups.
Dear Dikran Dikranjan ,
To reply to your question, one needs to know at least the following concepts: 1.Topological entropy. 2.Continuous endomorphism. 3.Locally compact. 4.Connected group. My Ph.D. is in differential topology, and to understand your request, I need to read your publications about the topic of the question as https://arxiv.org/pdf/1905.09516,
it is an excellent article and requires a lot of time to understand all results. I think the current problem is considered as a conjecture in one of them. It seems that the answer is not easy. Do you think that RG is the right platform to post your question in this complicated form? In my opinion, you can divide your question as a series of questions, that may attract more followers, for example: 1. What is your favorite connected group or ( Lie group)? 2. Are there any applications about Lie groups? 3. What are the different types of topological entropy? Etc. Similar simple questions may attract researchers to follow and read your work. Best regards
Dear Peter Breuer ,
For topological entropy ,
see
https://en.wikipedia.org/wiki/Topological_entropy
www.scholarpedia.org/article/Topological_entropy
or
https://pdfs.semanticscholar.org/1cbf/a8a8bb67ec20a8258cdb4d7ad8abfe25e19e.pdf.
Connected is related to the relative topology.
Best regards
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