Limits on Supreme Court's powers

No. The opinion by the Supreme Court of the United States in Thompson did not address this question. It is simply a case where certiorari was filed despite there being no state appellate process of which the petitioner could first avail himself. But no one argued that he couldn’t file for certiorari, and thus we do not know if the Court would hold that certiorari must be allowed by Court rule or federal statute.

See the opinion in Thompson v. City of Louisville.

McKane v. Durston (1894), 153 U.S. 684. State courts have the right to grant appellate review at their discretion.

Abney v. United States (1977), 431 U.S. 651, 656. The right to appeal in a criminal case is established by statute.

Halbert v. Michigan (2005), 125 S.Ct. 2582. A defendant has the right to appointed counsel in order to pursue leave to appeal from a guilty or no contest plea.

State v. Butts (1996), 112 Ohio App. 3d 683. Following a guilty verdict, the defendant agreed not to appeal in exchange for dismissal of a specification alleging a prior aggravated felony conviction. The right to appeal may be voluntarily waived.
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State v. Ashworth* (1999), 85 Ohio St. 3d 56, 64. “…[A] defendant cannot waive this court’s review of his death sentence, though he can waive review of his conviction.”

And see the opening paragraph and citations here: http://digitalcommons.law.yale.edu/cgi/viewcontent.cgi?article=3087&context=fss_papers

Marbury v Madison itself is a precedent supporting the claim I made before. I don’t find anything in it which suggests that only the Supreme Court has the power to scrutinise statutes for constitutionality and strike them down if found unconstitutional; it assigns that function to the judiciary in general. And the language of the case makes it very clear that this function is derived directly from the constitution.

Conclusion? If Congress took away appellate jurisdiction from the Supreme Court, the lower courts would still continue to exercise judicial review over statutes for constitutionality. Nothing short of a constitutional amendment could abolish that.

US Marshals, while part of the executive branch, work for the judicial branch.

While they are chiefly used for the transport of federal prisoners and apprehension of federal fugitives (like this documentary) are there any regulations preventing SCOTUS or the lower federal courts from using them to enforce their decisions?

No. I realize that it is a subtle distinction, and a difficult one to understand without an actual effort, but the court did NOT write law in cases such as Miranda v. Arizona. What they did, was they declared that the Constitution ALREADY REQUIRED that certain things be done. That is not at all the same as creating a new law.

SCOTUS even has its own police force, but I’ve never heard of them being used to enforce the judgments of the court. it’s just too small a group of officers for that, and it’s not their job.

https://en.wikipedia.org/wiki/Supreme_Court_Police

Does this imply that the President and Congress, acting independently of each other, do not have such a recourse? I’m thinking of course of the current President’s attempts to bar immigration from certain countries. If SCOTUS blocks him for instance is Trump powerless without Congress to do anything about it?

Respectfully, I think you’re misreading Thompson. The Court expressly stated that the basis for cert. was the existence of a constitutional issue. Nothing more, end of story. And that’s my point – the existence of a constitutional question is, in and of itself, a sufficient basis for Supreme Court review. The posture and procedural history thus don’t matter.

You’re postulating that there might be a future case in which different arguments would cause the Court to add additional criteria to the availability of cert., but it’s always possible that a future case will change the law, and that doesn’t stop us from treating the Court’s current construction of the law as good law.

As I see it, after Thompson (and Cooper), the burden is now on you. Show us case law or any other authority saying that there can exist a constitutional question that the Supreme Court cannot resolve if it chooses.